Document [original]

Geometric Complexity Theory

and Orbit Closures of Homogeneous Forms

vorgelegt von

Diplom-Mathematiker

Jesko Hüttenhain

aus Siegen

Von der Fakultät II - Mathematik und Naturwissenschaften

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften

Dr. rer. nat.

genehmigte Dissertation

Promotionsausschuss

Vorsitzender: Prof. Dr. Jörg Liesen

Gutachter : Prof. Dr. Peter Bürgisser

Gutachter : Prof. Dr. Giorgio Ottaviani

Tag der wissenschaftlichen Aussprache: 3. Juli 2017

Berlin 2017

Niemals aufgeben, niemals kapitulieren.

— Peter Quincy Taggart

Deutsche Einleitung

Das P-NP-Problem gehört zu den fundamentalsten und faszinierensten Problemen der

heutigen Mathematik. Es hat umfangreiche Bedeutung für praktische Anwendungen

und ist gleichermaßen eine grundsätzliche Frage über die Natur der Mathematik an

sich. Wäre unerwartet P=NP, so könnte etwa ein Computer effizient bestimmen,

ob eine mathematische Aussage wahr oder falsch ist. Seit die Frage 1971 von Cook

[Coo71] gestellt wurde, scheinen wir einer Antwort jedoch nicht nennenswert näher

gekommen zu sein. Der größte Fortschritt ist das ernüchternde Resultat von Razborov

und Rudich [RR97], dass es keine „natürlichen“ Beweise für P=NP geben kann, siehe

deren Arbeit für eine Definition und Details.

Peter Bürgisser hat gezeigt [Bü00], dass unter der verallgemeinerten Riemann-

Hypothese die nicht-uniforme Version von P=NP eine Vermutung von Valiant im-

pliziert, welche weithin als ein algebraisches Analogon betrachtet wird. Diese Vermu-

tung ist ein ebenso offenes Problem wie das ursprüngliche, doch es gibt die Hoffnung,

dass die zusätzliche algebraische Struktur mehr Ansatzpunkte liefert. Wir geben einen

kurzen Überblick über die zugrundeliegende Theorie in Kapitel 1: Die besagte Ver-

mutung von Valiant (Vermutung 1.4.5) ist die zentrale Motivation für die hier vorge-

stellten Forschungsergebnisse.

In Kapitel 2verschärfen wir ein Resultat von Valiant indem wir zeigen, dass sich

jedes ganzzahlige Polynom stets als Determinante einer Matrix schreiben lässt, de-

ren Einträge nur Variablen, Nullen und Einsen sind. The Größe der kleinsten solchen

Matrix ist ein sinnvolles Komplexitätsmaß, welches sich rein kombinatorisch untersu-

chen lässt. Als Anwendung beweisen wir untere Schranken in kleinen Fällen durch

Computerberechnung.

Von hier an widmen wir uns einem Ansatz zum Beweis von Valiant’s Vermutung,

welcher 2001 von Mulmuley und Sohoni vorgestellt wurde und den Titel „Geometric

Complexity Theory“ trägt, oder auch kurz GCT. Valiants Ergebnisse werden hier in

Aussagen der Algebraischen Geometrie und Darstellungstheorie übersetzt, um da-

durch die Probleme vorheriger Ansätze zu vermeiden [For09]. Eine Zusammenfas-

sung dieses interessanten Übersetzungsprozesses ist in Kapitel 3zu finden.

iii

Um Komplexitätsklassen voneinander zu trennen, muss bewiesen werden, dass

kein effizienter Algorithmus für ein mutmaßlich schweres Problem existiert. GCT lie-

fert ein Kriterium, wonach die Existenz gewisser ganzzahliger Vektoren schon diese

Trennung impliziert. Solche Vektoren nennen wir auch Obstruktionen. Die Hoffnung

von Mulmuley und Sohoni war, dass man sich für den Beweis von Valiants Vermu-

tung auf eine bestimmte Art von Obstruktionen beschränken kann, doch wir konn-

ten bereits 2015 bemerken, dass es starke Anzeichen dagegen gibt. Wir stellen diese

Ergebnisse in Kapitel 4vor. Erst kürzlich haben Bürgisser, Ikenmeyer und Panova

schlussendlich gezeigt, dass diese sogenannten occurrence obstructions nicht ausrei-

chen, um Valiants Vermutung zu beweisen.

Im zweiten Teil der Arbeit beschäftigen wir uns näher und in größerer Allgemein-

heit mit einem der zentralen Konzepte von GCT: Wir betrachten die Wirkung der GLn

auf dem Raum der homogenen, n-variaten Polynome vom Grad ddurch Verkettung

von rechts und beobachten, dass alle Elemente einer Bahn im Wesentlichen die gleiche

Berechnungskomplexität haben. Man ordnet einem Polynom Pdas kleinste dzu, so

dass Pim Bahnabschluss des d×d– Determinantenpolynoms liegt. Diese Kennzahl

ist dann äquivalent zu Valiants ursprünglichem Komplexitätsmaß für Polynome, so-

fern auch Approximationen zugelassen werden. Die Geometrie des Bahnabschlusses

der Determinante ist wenig verstanden, sogar für kleine Werte von d. Wir können al-

lerdings die im Rand auftretenden Komponenten für d=3 vollständig klassifizieren.

Die benötigten algebraischen und geometrischen Werkzeuge werden im einfüh-

renden Kapitel 5vorgestellt. Bahnabschlüsse von homogenen Formen sind stets al-

gebraische Varietäten mit der oben genannten GLn-Wirkung und werden klassisch

in der geometrischen Invariantentheorie studiert. Neben der Determinante beschäfti-

gen wir uns in den darauffolgenden Kapiteln auch mit den Bahnabschlüssen anderer

homogener Formen.

In Kapitel 6behandeln wir das allgemeine Monom x1···xd, dessen Relevanz für

GCT daher stammt, dass es auch als Einschränkung des Determinantenpolynoms auf

Diagonalmatrizen verstanden werden kann. Das Studium seines Bahnabschlusses ist

beispielsweise das zentrale Hilfsmittel in Kapitel 4. Wir bemerken in diesem Kapitel,

dass das Monom die seltene Eigenschaft hat, dass jedes Polynom in seinem Bahnab-

schluss lediglich das Ergebnis einer Variablensubstitution ist. Eine Klassifikation aller

Polynome mit dieser Eigenschaft bleibt offen, obwohl wir einige Fragen im Hinblick

darauf beantworten können.

Wir stellen für die beiden abschließenden Kapitel noch weitere Techniken in Kapi-

tel 7vor. Maßgeblich ist eine obere Schranke für die Anzahl irreduzibler Komponen-

ten des Randes eines Bahnabschlusses. Die erste Anwendung dieser Aussage liefert

eine Beschreibung des Randes für det3in Kapitel 8. Wir bestimmen hier auch den

Stabilisator der Determinante einer allgemeinen, spurlosen Matrix und können so

schlussfolgern, dass der Bahnabschluss dieses Polynoms stets eine Komponente im

Rand des Bahnabschlusses der Determinante ist.

Das letzte Kapitel enthält bisher unveröffentlichte Ergebnisse über das allgemeine

Binom x1···xd+y1···yd. Wie auch das Monom ist das Binom im Bahnabschluss

von detdenthalten. Ein solides Verständnis dieser Familie homogener Formen sollte

daher Voraussetzung dafür sein, den Bahnabschluss des Determinantenpolynoms im

Allgemeinen zu studieren. Die hier aufkommenden geometrischen Fragen sind bereits

deutlich komplexer als im Fall des Monoms: Wir können zwei Komponenten des

Randes im Detail beschreiben, müssen jedoch eine subtile Frage offen lassen. Sofern

diese sich positiv beantworten lässt, erhalten wir jedoch so bereits eine vollständige

Beschreibung des Randes.

Contents

Acknowledgements 1

Introduction 3

I Geometric Complexity Theory 7

1 Algebraic Complexity Theory 9

1.1 Arithmetic Circuits ............................... 9

1.2 The Classes VP and P ............................. 12

1.3 Reduction and Completeness ......................... 13

1.4 The Classes VNP and NP ........................... 14

1.5 Determinant Versus Permanent ........................ 16

2 Binary Determinantal Complexity 21

2.1 The Cost of Computing Integers ....................... 22

2.2 Lower Bounds .................................. 26

2.3 Uniqueness of Grenet’s construction in the 7 ×7 case ........... 29

2.4 Algebraic Complexity Classes ......................... 30

2.5 Graph Constructions for Polynomials .................... 31

3 Geometric Complexity Theory 35

3.1 Orbit and Orbit Closure as Complexity Measure .............. 35

3.2 Border Complexity ............................... 37

3.3 The Flip via Obstructions ........................... 39

3.4 Orbit and Orbit Closure ............................ 41

4 Occurrence Obstructions 45

4.1 Weight Semigroups ............................... 45

4.2 Saturations of Weight Semigroups of Varieties ............... 47

4.3 Proof of Main Results ............................. 49

vii

II Orbit Closures of Homogeneous Forms 55

5 Preliminaries 57

5.1 Conciseness ................................... 58

5.2 Grading of Coordinate Rings and Projectivization ............. 60

5.3 Rational Orbit Map ............................... 61

6 Closed Forms 65

6.1 A Sufficient Criterion .............................. 65

6.2 Normalizations of Orbit Closures ....................... 67

6.3 Proof of Main Theorem ............................ 71

7 Techniques for Boundary Classification 77

7.1 Approximating Degenerations ........................ 77

7.2 The Lie Algebra Action ............................. 80

7.3 Resolving the Rational Orbit Map ...................... 82

8 The 3 by 3 Determinant Polynomial 89

8.1 Construction of Two Components of the Boundary ............ 90

8.2 There Are Only Two Components ...................... 91

8.3 The Traceless Determinant ........................... 96

8.4 The Boundary of the 4 ×4 Determinant ................... 102

9 The Binomial 107

9.1 Stabilizer and Maximal Linear Subspaces .................. 108

9.2 The First Boundary Component ....................... 110

9.3 The Second Boundary Component ...................... 116

9.4 The Indeterminacy Locus ........................... 118

III Appendix 123

A Algebraic Groups and Representation Theory 125

A.1 Algebraic Semigroups and Groups ...................... 125

A.2 Representation Theory of Reductive Groups ................ 130

A.3 Polynomial Representations .......................... 139

Bibliography 141

List of Symbols 149

Index 153

viii

Acknowledgements

I had the immense privilege to work in the most comfortable scientific environment

imaginable, having both the freedom to pursue my interests and the support of my

advisor, Peter Bürgisser. I want to thank him heartily for providing this and also

for the many things that I learned from him. Researching under these great con-

ditions was also made possible by the grants BU 1371/2-2 and BU 1371/3-2 of the

Deutsche Forschungsgemeinschaft. Their support is highly appreciated. In late 2014 I

participated in the program on Algorithms and Complexity in Algebraic Geometry at the

Simons Institute with great benefit. I am very grateful for this opportunity and extend

my thanks to the organizers Peter Bürgisser, Joseph Landsberg, Ketan Mulmuley and

Bernd Sturmfels.

I also had the pleasure to enjoy the company of extraordinary colleagues, two of

which are also my coauthors. I thank Christian Ikenmeyer and Pierre Lairez for the

confusion we shared, the time we spent pondering and the insights we could claim

together.

My parents have given me more than words could ever say, and my gratitude for

all their love and support is eternal.

Ever since a memorable chess game in late 2004, I am part of a friendship that

particularly sustained me throughout the years. I want to thank Nikolai Nowaczyk

for sharing it with me – across time, space, and the fall of empires.

Introduction

The Pvs. NP problem is among the deepest and most intriguing questions of math-

ematics today. While having a multitude of implications for practical applications, it

also carries fundamental questions in its wake, about the nature of mathematics itself.

If for example P=NP would hold unexpectedly, then we could program a machine

to efficiently determine the truth of mathematical statements. Introduced in 1971 by

Cook [Coo71], the problem is already 46 years old and we seem as far from the so-

lution as ever. The most notable progress so far is the sobering result by Razborov

and Rudich [RR97] that no “natural” proof for P=NP exists, see their work for a

definition and details.

Peter Bürgisser has shown [Bü00] that the nonuniform version of P=NP im-

plies, under the generalized Riemann hypothesis, a conjecture by Valiant which is

widely considered an algebraic analogue. It remains as unresolved as the former,

even though the additional algebraic structure involved is believed to provide better

points of vantage. We give a brief review of the underlying theory in Chapter 1– It

is the said conjecture by Valiant (Conjecture 1.4.5) that serves as motivation for the

research presented herein.

In Chapter 2, we slightly strengthen a result by Valiant: The core observation is

that integer polynomials can always be written as the determinant of a matrix whose

entries are variables, zeros and ones. The size of the smallest such matrix then gives a

reasonable complexity measure which at the same time is accessible to combinatorics.

As an application, we can provide lower bounds in small cases by computational

methods.

From there on, we are concerned with a recent approach to Valiant’s Conjecture

known as Geometric Complexity Theory, or GCT for short. Introduced in 2001 by

Mulmuley and Sohoni, it avoids the difficulties of many previous attempts [For09] by

translating Valiant’s results to statements in algebraic geometry and representation

theory. An outline of this enticing transition is given in Chapter 3.

Where complexity theory is clasically concerned with proving the nonexistence of

good algorithms for supposedly hard problems, GCT provides a criterion whereby the

existence of certain integer vectors implies a separation of complexity classes. We refer

to the vectors in question simply as obstructions. It was the hope of Mulmuley and

Sohoni that the search could be further restricted to a particular kind of obstructions,

but we already observed in 2015 that this appears unlikely – we present these results

in Chapter 4. Quite recently in 2016 it was shown by Bürgisser, Ikenmeyer, and

Panova that indeed these so-called occurrence obstructions do not suffice to prove

Valiant’s Conjecture, representing a bitter setback for the programme.

For the second part of the thesis, we treat a central topic of GCT in more detail and

generality: One can measure the approximate complexity of polynomials by studying

the closure of their orbit under the action of a general linear group by precomposi-

tion. The smallest dfor which a polynomial appears in the orbit closure of the d×d

determinant polynomial is equivalent to Valiant’s original measure for its complexity,

if approximations are permitted. The geometry of the determinant orbit closure is

little understood even for small values of d, but we can give a classification of the

components that appear in the boundary for d=3.

The required toolbox of geometric and algebraic techniques is introduced in the

preliminary Chapter 5. Quite generally, the orbit closure of a homogeneous form is an

algebraic variety with a GLn-action which is both intuitive and yet incredibly intrigu-

ing, and is an object of study to the beautiful fields of classical geometric invariant

theory and birational geometry. In the subsequent chapters we study this problem for

other homogeneous forms than the determinant.

Chapter 6deals with a polynomial that has appeared in the context of GCT before,

namely the universal monomial x1···xd. Its orbit closure is contained in the orbit

closure of detdand was instrumental in proving the results of Chapter 4, for example.

It also has the remarkable and rare property that every polynomial in its orbit closure

is the result of a variable substitution. A classification of all polynomials with this

property remains open, but we both answer and pose several questions to advance it.

In Chapter 7, we introduce additional techniques used in the two subsequent chap-

ters: We bound the number of irreducible components of the orbit closure boundary

by the number of smooth blowups required to resolve the indeterminacy of a related

rational map.

The first application of this technique yields a classification of the boundary of det3

in Chapter 8. Here, we also describe the stabilizer group of the determinant of a

generic traceless matrix and conclude that the orbit closure of this polynomial is

always a codimension one component of the boundary of the orbit closure of the

determinant.

The final chapter contains unpublished work on the universal sum of two mono-

mials, the binomial x1···xd+y1···yd. Like the monomial, the binomial is contained

in the orbit closure of detd. A firm understanding of this polynomial should therefore

precede the study of general determinantal expressions. The binomial already gives

rise to a significantly more involved geometry than the monomial. We can describe

two components of the boundary in detail, but must leave a subtle question open. If

said question could be answered affirmatively however, these two components con-

stitute the entire boundary of the orbit closure of the binomial.

Part I

Geometric Complexity Theory

Chapter 1

Algebraic Complexity Theory

The following is only the briefest of summaries: Complexity theory analyzes the

complexity of solving problems. Problems have instances of different sizes which can

be solved by algorithms. Denoting by t(m)the minimum number of steps performed

by any algorithm that solves all instances of size m, the complexity of a problem is

the function t:N→N. The definitions of “problem”,“algorithm” and related notions

is encompassed by a model of computation. The classical model of computation is

the Turing machine, which mimics our present-day computers. Liberally quoting the

famous Church-Turing thesis, any problem that we consider computationally solvable

is solvable by a Turing machine. This generality comes at a price, however: The Turing

model provides very little mathematical structure to be exploited.

In this chapter, we explore the algebraic model, where a problem is given as a

family of polynomial functions and the goal is quite simply to evaluate them.

1.1 Arithmetic Circuits

Figure 1.1.1: x2+xy

The algorithms in algebraic complexity theory are arith-

metic circuits. An arithmetic circuit is a schematic repre-

sentation of a way to compute a polynomial: Figure 1.1.1

shows a circuit computing x2+xy ∈Z[x,y]. In general, an

arithmetic circuit is a directed, acyclic graph where at every

vertex, a polynomial is being computed. Vertices with no

ingoing edges contain constants or variables and vertices

with exactly two ingoing edges are either labeled with the

symbol “+” (Plus) or “×” (Times), computing the sum or

the product of their input, respectively. This concept can be generalized to the notion

of a circuit, which performs computation in an algebraic structure by associating to

any vertex with kingoing edges some k-ary operation.

1.1.1 Definition. Let Rbe a commutative ring and R[x]the polynomial ring over R

in a countably infinite set of variables x. An arithmetic circuit over Ris a directed

acyclic graph Cwith vertex labels, subject to the following conditions:

(1) The vertices with no incoming edges are labelled with elements of R∪x. Any

such vertex is called an input gate.

(2) Since Cis acyclic, every vertex of Chas a well-defined depth, which is the length

of a longest path from an input gate to it. Any vertex of positive depth has exactly

two incoming edges and is labelled with an element of {+,×}. Any such vertex

is also called a computation gate.

We define P(v)∈R[x]for every vertex vrecursively by depth, as follows: If vis an

input gate, P(v)is defined to be the label of v. Otherwise, let ∗v∈{+,×}be the label

of vand denote by uand wthe source vertices of the two incoming edges of v. Then,

we can define P(v):=P(u)∗vP(w).

The circuit Ccomputes an element P∈R[x]if there is a vertex vwith P=P(v).

The size of Cis the number of computation gates, denoted by|C|.

1.1.2 Example. In Figure 1.1.2, we give an example of an arithmetic circuit over C

which computes the equation of an affine elliptic curve in the variables{x,y}.

××

−1xy

Figure 1.1.2: An arithmetic circuit computing y2+xy −x3−1.

1.1.3 Definition. Let Rbe a commutative ring. The (circuit) complexity of a poly-

nomial Pwith coefficients in Ris the minimum size of an arithmetic circuit over R

which computes P. We denote this number by ccR(P).

It should be emphasized that complexity theory does not study the complexity of

single polynomials. Instead, the object of study are families of polynomials. This is

the main reason why it is convenient to work with infinitely many variables.

1.1.4 Example. Consider the polynomial Pm:=x2

1+···+x2

m∈R[x]for m∈N. We

study the map N→Ngiven by m↦→ ccR(Pm). We claim that 1

2m≤ccR(Pm)≤2m,

which is interpreted by saying that the complexity of computing a Euclidean norm is

linear in the input size. It is easy to see that the circuit

···

× × × ··· ×

x1x2x3··· xm

computes Pmwith 2m−1 computation gates. Since Pmis supported on mvariables,

each of which needs to be connected to some computation gate in a circuit comput-

ing Pm, we can see that ccR(Pm)≥1

2mbecause at most two variables can be connected

to the same gate. With a more refined argument, one can actually show ccR(Pm)≥m

in an even stronger model of computation [BCS97, Example 6.1].

1.1.5 Notation. We fix a commutative ring Rand a countably infinite set xof variables.

A polynomial will always be an element of R[x]unless stated explicitly otherwise. We

will also write P∈R[x1, . . . , xn]for a polynomial in nvariables and implicitly identify

the xiwith certain elements of x.

The elements of xare not sequentially numbered or named in any way, a priori.

We do not assume any ordering on x. We may often write x1, . . . , xr∈x, but we

may also write a,b,c∈xor y∈x.

This way, we always operate in the ring R[x], regardless of what (finite) number of

variables we require. Finally, we write cc(P)instead of ccR(P)unless we want to put

emphasis on the ring R.

Remark. In Chapter 2,Rwill be equal to Zand starting with Chapter 3,Rwill be the

field of complex numbers.

1.2 The Classes VP and P

Only for very few and rather simple families (Pm)m∈Nof polynomials, we can deter-

mine the function m↦→ cc(Pm)“explicitly”. As is often the case in complexity theory,

we restrict instead to the classification of such functions by their asymptotic rate of

growth. Even this task turns out to be quite challenging.

1.2.1 Definition. A function t:N→Nis called polynomially bounded if there is a

polynomial p∈Z[x]and m0∈Nsuch that ∀m≥m0:t(m)≤p(m). We define poly

as the set of all polynomially bounded functions N→N.

Remark. We will often write t(m)∈poly(m)instead of t∈poly, for example we

would write m3∈poly(m)instead of giving the function m↦→ m3a name. If a

numeric quantity t(m)∈Nis associated to each element of a family P= (Pm)m∈N,

we say that Phas (or admits) polynomially many of said quantity to express that

t∈poly.

It turns out that it is not desirable to study arbitrary families of polynomials.

For example, the value x2mfor x∈Zcannot be computed by a Turing machine in

polynomial time simply because the output is too large, but the circuit

××

···

××

computes it with only mcomputation gates.

1.2.2 Definition. We say that Pis a p-family if P= (Pm)m∈Nis a family of polyno-

mials with deg(Pm)∈poly(m). The complexity class VP is defined to be the set of all

p-families Pwith cc(Pm)∈poly(m).

Remark. Although it is not visible in the notation, this definition depends on the coef-

ficient ring R.

VP is the circuit analogue of the complexity class P. The “V” in its name stands for

the name of its inventor Valiant, who introduced these notions in [Val79a].

1.2.1 The Class P

Classical complexity theory deals with families of boolean functions Fm

2→F2. A

family Bm:Fm

2→F2is in the complexity class Piff there exists a Turing machine

which computes Bmand requires polynomially many steps to do so. We do not give

the formal definition of a Turing machine, but it may be thought of as a computer

algorithm, where a step is a physically atomic operation for the machine.

The class Phas the nonuniform analogue P/ poly which has a description in terms

of circuits: Nonuniformity means that we are not asking for a single algorithm to

work on all input sizes, but we allow different algorithms for different input sizes. In

the language of circuits, a family Bm:Fm

2→F2of boolean functions is in P/ poly if

and only if ccF2(Bm)∈poly(m). Note that by polynomial interpolation, any function

2→F2is given by some polynomial in F2[x1, . . . , xm]⊆F2[x].

1.3 Reduction and Completeness

An important concept in boolean as well as algebraic complexity theory is reduction.

Informally put, a problem Qcan be reduced to a problem Pif one can produce an

algorithm for Qfrom an algorithm for Pwithout measurably increasing the runtime.

This is only a vague description, of course. We will give the precise definition for the

algebraic model after introducing some notation.

1.3.1 Definition. Denote by N(x)the x-indexed sequences α= (αx)x∈xof natural

numbers with αx=0 for only finitely many x∈x. By definition, a polynomial

P∈R[x]is a map P:N(x)→Rwhich we denote α↦→ Pαand require Pα=0 for only

finitely many α∈N(x). One then writes

P=∑

α∈N(x)

Pα·∏

x∈x

xαx.

We define the support of Pto be the set of variables that occur in this expression, i.e.

supp(P):={x∈x⏐⏐⏐∃α∈N(x):αx=0∧Pα=0}.

Note that supp(P)is always a finite set. For P∈R[x]with supp(P) = {x1, . . . , xn},

the expression for Pbecomes the familiar P=∑α∈NnPα·∏n

i=1xαi

1.3.2 Example. Let x,y,z∈x. For Q=x2y+z3, we have supp(Q) = {x,y,z}. The

polynomial P=x2y+y3can also be viewed as an element of C[x,y,z]but we have

supp(P) ={x,y}because zdoes not occur in P.

1.3.3 Remark. If P∈VP, then |supp(Pm)|∈poly(m). In other words, any family

in VP requires only polynomially many variables. Indeed, in a circuit of minimum

size computing Pm, every input gate containing a variable is connected to at least one

computation gate and each computation gate can be connected to at most two input

gates. Since there are only polynomially many computation gates, there can only be

polynomially many variables in a (nonredundant) expression for Pm.

We next define a partial order “≤” on R[x]where P≤Qholds if Pcan be ob-

tained from Qby variable substitution. For example, P≤Qholds in Example 1.3.2

because Parises from Qby substituting zfor y. This is formalized in the following

Definition 1.3.4:

1.3.4 Definition. For S⊆xand a map σ:S→R[x], we extend σto a map x→R[x]

by the identity. We then denote by Pσthe image of Punder the unique R-algebra

homomorphism R[x]→R[x]which maps x↦→ σ(x)for all x∈x.

A polynomial Pis called a projection of another polynomial Qif there is some

S⊆supp(Q)and a map σ:S→x∪Rsuch that P=Qσ. We denote this by P≤RQ.

The definition extends to families as follows: A p-family Pis a p-projection of another

p-family Q, in symbols P≤RQ, if there is some t∈poly such that Pm≤Qt(m)for all

m∈N. We will write P≤Qin both cases if there is no ambiguity concerning R.

1.3.5 Example. Let x,y,z∈xand Q:=y2z+xyz −x3−z3. Let σ:x→R[x]be the

identity everywhere except for σ(z) = 1. Then, one obtains Qσ=y2+xy −x3−1.

1.3.6 Example. Let ai∈xand xi∈xbe distinct variables for all i∈N. For d∈N, we

consider the family Q= (Qd)d∈Ngiven by

Qd:=a0+a1x1+. . . +adxd

with supp(Qd) ={a0, . . . , ad,x1, . . . , xd}. Any p-family P∈VP of affine linear polyno-

mials satisfies P≤Q. Indeed, this follows because t(m):=|supp(Pm)|is polynomially

bounded by Remark 1.3.3.

1.4 The Classes VNP and NP

Valiant introduced the class VNP as an analogue of the class NP, but we will give a

definition of the class VNP first because our focus is on the algebraic model. We will

then discuss the relation to NP.

1.4.1 Definition. Let Pbe a p-family. Then P∈VNP if and only if there exists a family

Q∈VP and a sequence (Cm)m∈Nwith Cm⊆supp(Qm)such that Pis a projection of

the family ˜

Q= ( ˜

Qm)m∈N, which is defined as

Qm:=∑

σ:Cm→{0,1}

Qσ

m·∏

x∈Cm

σ(x)=1

Remark. We use here the original definition from [Val79a, p. 252] because we can later

make the analogy to classical complexity classes more clear. In subsequent literature,

the definition appears in seemingly weaker, but equivalent form [Bü00, p. 5].

Remark. It follows from the definition that VP ⊆VNP: For P∈VP, choose Q=Pand

Cm:=∅for all m∈N. By Definition 1.3.1, the empty map σ:∅→{0,1}satisfies

Pσ

m=Pmfor all m∈N.

Remark. Note that for P∈VNP, we again have |supp(Pm)|∈poly(m). Indeed, in

Definition 1.4.1 we know that |supp(Qm)|∈poly(m)by Remark 1.3.3. Furthermore,

supp(˜

Qm)⊆supp(Qm)and and since Pm≤˜

Qm, we have

|supp(Pm)|≤⏐⏐supp(˜

Qm)⏐⏐≤|supp(Qm)|∈poly(m).

1.4.1 The Class NP

We will now explain and motivate this definition by comparing Valiant’s class with

the classical one. A family Bm:Fm

2→F2is in NP if and only if there are functions

tm∈poly(m)and Cm:Fm

2×Ftm

2→F2such that

• the family (Cm)m∈Nis in Pand

•∀m∈N:∀σ∈Fm

2:(Bm(σ) = 1)⇔(∃c∈Ftm

2:Cm(σ,c) = 1).

In other words, it might not be easy to decide whether Bm(σ) = 1, but it is easy to

confirm that Bm(σ) = 1 if given a valid certificate c∈Ftm

Again, there is a relevant related complexity class known as #P. It contains families

of maps Fm

2→N. Such a family Pm:Fm

2→Nis in #Pif and only if there exists a

family (Bm)m∈N∈NP which has a certificate function (Qm)m∈N∈Psuch that Pm

counts the number of certificates, i.e.,

Pm(σ) =⏐⏐⏐{c∈Ftm

2⏐⏐⏐Qm(σ,c) = 1}⏐⏐⏐.

Note that Bm(σ) = 1 if and only if Pm(σ)>0.

If we interpret Qm∈F2[x,y]where x={x1, . . . , xm}and y={y1, . . . , ytm}, then

Qm:=∑

σ:y→{0,1}

Qσ

m·

∏

k=1

σ(yk)=1

is an element of VNP over F2by Definition 1.4.1 and for any σ∈Fm

2=Fx

2, i.e.,

viewing σas a map σ:x→{0,1}=F2, we have ˜

Qσ

m=0 if and only if Bm(σ) = 0.

More precisely, ˜

Qσ

mhas exactly Pm(σ)monomials and the monomial

∏tm

k=1

ck=1yk

occurs in ˜

Qσ

mif and only if c∈Ftm

2is a certificate with Qm(σ,c) = 1.

1.4.2 Completeness of the Permanent

1.4.2 Definition. Let Cbe some class of p-families, like VP or VNP. A p-family Pis

called C-complete if P∈ C and ∀Q∈ C:Q≤P.

Two completeness results by Valiant make the connection between #Pand VNP

even more tangible. They both concern the permanent polynomial family. The per-

manent of an m×mmatrix (xij)1≤i,j≤mis defined as

perm:=∑

π∈Sm

∏

i=1

xi,π(i), (1)

where Smdenotes the symmetric group on msymbols, i.e., the group of all set auto-

morphisms of [m]:={1, . . . , m}. We will consider per = (perm)m∈Nas a family of

polynomials in the variables xij.

1.4.3 Theorem ([Val79b]). The problem of computing the permanent of a binary ma-

trix (one where every entry is either 0 or 1) is #P-complete.

1.4.4 Theorem ([Val79a]). The permanent family is VNP-complete over any field of

characteristic different from 2.

By a formula of Ryser [Rys63], we know that permcan be computed by a circuit

of size m·2m, but no significant improvement over this is known. This leads to the

following conjecture:

1.4.5 Conjecture (Valiant’s Hypothesis). The inclusion VP ⊆VNP is strict.

Given any VP-complete family P, one could prove VP =VNP by showing that the

permanent is not a p-projection of P.

1.5 Determinant Versus Permanent

The definition of the permanent (1) is quite similar to the familiar definition of the

determinant family

detd:=∑

π∈Sd

sgn(π)·

∏

i=1

xi,π(i).

It is known [MV97] that detdcan be computed by a circuit of size O(d6)– this is not

immediately obvious because the classical procedure of Gaussian elimination per-

forms divisions: Arithmetic circuits only allow for multiplication and addition.

Unfortunately, the determinant is not known to be VP-complete, nor is it thought

to be. However, it is complete for a subclass of VP which we will study next.

1.5.1 Classes for the Determinant

The class of polynomials for which the determinant family is complete can be defined

in several different ways. We will not go into detail and instead refer to [MP08;Tod92].

However, we give one definition:

An arithmetic circuit Cis weakly skew if every vertex with label “×” has one

incoming edge which is a bridge, i.e., removing this edge increases the number of

connected components of C.

1.5.1 Example. Recall the circuit computing y2+xy −x3−1 from Example 1.1.2. It

is not weakly skew, the condition is for example violated at the topmost multiplica-

tion gate computing xy −x3−1. However, the circuit can be made weakly skew by

duplicating some of the input gates: See Figure 1.5.1. At each multiplication gate, the

input edge which is a bridge has been highlighted.

××

−1

x x x y y

Figure 1.5.1: A weakly skew arithmetic circuit computing y2+xy −x3−1.

Denote by ccws (P)the minimum number dsuch that Pcan be computed by a weakly

skew circuit of size d.VPws consists of all p-families Psuch that ccws (Pm)∈poly(m).

1.5.2 Theorem ([Tod92],[MP08, Lemma 6]). The determinant family is VPws-complete

and for any P∈R[x]with d:=ccws (P), we have P≤detd+1.

Clearly, VPws ⊆VP and therefore, if Valiant’s Hypothesis (Conjecture 1.4.5) holds,

the inclusion VPws ⊆VNP must also be strict. Since the determinant is VPws-complete

and the permanent is VNP-complete by Theorem 1.4.4,VPws =VNP is equivalent to

the following:

1.5.3 Conjecture (Permanent vs. Determinant). The permanent polynomial family is

not a p-projection of the determinant polynomial family.

1.5.2 Determinantal Complexity

In order to approach Permanent versus Determinant (Conjecture 1.5.3), we need to

study p-projections of the determinant. We will show that the determinant is uni-

versal, meaning that every polynomial is a projection of detdfor some d∈N. This

important insight gives rise to the definition of determinantal complexity: For a poly-

nomial Pwe define

dc(P):=min{d∈N|P≤detd}.

This allows us to reformulate Conjecture 1.5.3 simply as dc(perm)/∈poly(m). The

benefit of this reformulation is the fact that there are no more circuits involved in it

and methods from algebra directly apply.

1.5.4 Theorem ([Val79a, §2]). Let Rbe a commutative ring and P∈R[x1, . . . , xn]a

polynomial. Then, there is a natural number d∈Nsuch that Pis a projection of detd.

Remark. This highlights the importance of the determinant in a very fundamental

way. Quoting Valiant himself, “for the problem of finding a subexponential formula for a

polynomial when one exists, linear algebra is essentially the only technique in the sense that it

is always applicable.”

Proof. The proof is based on a graph construction which we will sketch here: Given

a matrix a= (aij)1≤i,j≤mwe can consider the labelled directed graph Gaon the set

of vertices {1, . . . , m}where the edge (i,j)has label aij. We treat edges with label 0

as nonexistent. For a directed graph with labels in R[x1, . . . , xn], we can reverse this

process and obtain a matrix from Gwhich we will refer to as its adjacency matrix. A

cycle cover of Gais a partition of Gainto vertex disjoint cycles. The permutations on the

set{1, . . . , m}are in bijection with the cycle covers of Ga. To each cycle we associate a

sign, which is −1 if the cycle has even length and 1 otherwise. To each cycle we then

associate a weight, which is its sign times the product of its edge labels. The weight

of a cycle cover is the product of the weights of its cycles. The determinant of ais

then, by definition, the sum of the weights of all cycle covers of Ga.

Given a polynomial P∈R[x1, . . . , xn], let k:=deg(P)be its (total) degree. A

product of k+1 constants and variables can produce any monomial that occurs in

P, so we can write P=∑r

i=1∏k

j=0Pij for some r∈Nand Pij ∈R∪{x1, . . . , xn}. For

example, consider the polynomial

P=x3+1−y2−xy = (x·x·x) + (1·1·1) + (−1·y·y) + (−1·x·y).

For this example, we used only 3 factors in each summand as opposed to the 4 factors

that the construction would yield for a general cubic polynomial.

−1

We construct a graph Has follows: For each

of the rsummands in the above representation

of P, consider a path with k+1 edges where the

edges of path iare labeled with the values pij

for 0 ≤j≤k.

We identify the start vertices of all these

paths and call it s, then we identify the end ver-

tices of all these paths and call it t. We obtain an acyclic graph with rpaths going

from sto t.

−1

4 1

−1

Figure 1.5.2: The Graph G

Let then Gbe the graph that arises from Hby

first adding loops with label 1 to all vertices ex-

cept sand tand then identifying sand tinto a

single vertex v.

By construction, every cycle in Gwhich is

not a loop contains the vertex vand all these cy-

cles have the same length. Therefore, the cycle

covers of Gare in bijection with the summands

∏k

j=0pij via their weight, up to a common sign.

It follows that the adjacency matrix aof Gsat-

isfies det(a) = ±P. If det(a) = −P, we can

achieve det(a) = Pby considering the block ma-

trix (a0

0−1)instead of a.

Remark. Treating the vertex vin our example as the 9-th vertex and numbering all

other vertices as in Figure 1.5.2, one can check that indeed

det ⎛

⎜

⎝

1y0000000

0 1 0 0 0 0 0 0 y

001x0 0 0 0 0

0 0 0 1 0 0 0 0 y

0 0 0 0 1 0 0 0 x

0 0 0 0 x1 0 0 0

000000101

000000110

−1 0 −100x010

⎞

⎟

⎠

=x3+1−y2−xy.

However, this procedure does not yield the optimal result. For example,

det (1 0 x0

0 1 0 y

0 0 1 x

x y y 1)=x3+1−y2−xy.

We used computational methods akin to the algorithm from the next chapter to com-

pute this representation.

Chapter 2

Binary Determinantal Complexity

This chapter contains the contents of the previously published work [HI16]. We only

consider polynomials with integer coefficients here, i.e., we assume R=Z.

Proving Valiant’s Hypothesis (Conjecture 1.4.5) amounts to bounding the growth

of dc(perm)superpolynomially from below. Lower bounds are a notoriously difficult

problem in complexity theory. On the other hand, finding upper bounds admits the

straightforward approach of constructing algorithms: In our case, an algorithm can

be described as a sequence of matrices Am∈(x∪Z)tm×tmsuch that det(Am) = perm.

Here, the numbers tm≥dc(perm)achieve equality if and only if the algorithm is

optimal. To get a better idea of how dc(perm)grows, it is therefore reasonable to

attempt the construction of good algorithms. The best one known so far is a graph

construction by Grenet [Gre11], see Section 2.5, with the following consequence:

2.0.1 Theorem. For every natural number mthere exists a matrix Aof size 2m−1

such that perm=det(A). Moreover, Acan be chosen such that the entries in Aare

only variables, zeros, and ones, but no other constants.

Theorem 2.0.1 gives rise to the following definition: We call a matrix whose entries

are only zeros, ones, or variables, a binary variable matrix. We will prove in Corol-

lary 2.1.3 that every polynomial Pwith integer coefficients can be written as the de-

terminant of a binary variable matrix and that the size is almost equal to dc(P), see

Proposition 2.1.2 for a precise statement. We then denote by bdc(P)the smallest d

such that Pcan be written as a determinant of an d×dbinary variable matrix. It is

called the binary determinantal complexity of P.

The complexity class of families (Pm)m∈Nwith polynomially bounded binary de-

terminantal complexity bdc(Pm)is exactly VP0

ws, the constant free version of VPws, see

Section 2.4 for definitions and proofs.

Theorem 2.0.1 shows that bdc(perm)≤2m−1. This upper bound is clearly sharp

for m=1 and for m=2, and we can also verify that it is sharp for m=3:

2.0.2 Theorem. bdc(per3) = 7.

We use a computer aided proof and enumeration of bipartite graphs in our study.

The binary determinantal complexity of permis now known to be exactly 2m−1 for

m∈{1,2,3}. Unfortunately, determining bdc(per4)is currently out of reach with our

methods.

The best known general lower bound is bdc(perm)≥dc(perm)≥m2

2due to

[MR04] in a stronger model of computation, see also [LMR13] for the same bound

in an even stronger model of computation. After the proof of Theorem 2.0.2 was

published, Alper, Bogart, and Velasco proved in [ABV15] that dc(per3) = 7, but un-

fortunately per4remains out of reach even with their methods.

2.1 The Cost of Computing Integers

The main purpose of this section is to prove that even though we only allow the

constants 0 and 1, all polynomials with integer coefficients can be obtained as the

determinant of a binary variable matrix, see Corollary 2.1.3. Moreover, the size of

the matrices is not much larger than had we allowed integer constants, see Propo-

sition 2.1.2. We use standard techniques from algebraic complexity theory, heavily

based on [Val79a], but a certain attention to the signs has to be taken.

In what follows, a digraph is always a finite directed graph which may possibly

have loops, but which has no parallel edges. We label the edges of a digraph by

polynomials. We will almost exclusively be concerned with digraphs whose labels

are only variables or the constant 1. Note that we consider only labeled digraphs.

A cycle cover of a digraph Gis a set of cycles in Gsuch that each vertex of G

is contained in exactly one of these cycles. If a cycle in Ghas iedges with labels

e1, . . . , ei, then its weight is defined as (−1)i−1·e1···ei. The weight of a cycle cover is

the product of the weights of its cycles. The value of Gis the polynomial that arises as

the sum over the weights of all cycle covers in G. We then define the directed adjacency

matrix A of a digraph G as the matrix whose entry Aij is the label of the edge (i,j)or

0 if that edge does not exist.

In what follows, we will often construct matrices as the directed adjacency matrices

of digraphs. The reason is the well-known observation that the value of a digraph G

equals the determinant of its directed adjacency matrix – see for example [Val79a].

As an intermediate step, we will often construct a binary algebraic branching pro-

gram: This is an acyclic digraph Γ= (Γ,s,t)where every edge is labeled by either 1 or

a variable. The digraph Γhas two distinguished vertices, the source s and the target t,

where shas no incoming and thas no outgoing edges. If an s-t-path in Γhas iedges

with labels e1, . . . , ei, then its path weight is defined as the value (−1)i−1·e1···ei.

The path value of Γis the polynomial that arises as the sum over the path weights of

all s-t-paths in Γ. We remark that this notion of weight differs from the literature by a

sign.

2.1.1 Proposition. For a nonzero constant c∈Z, there is a binary algebraic branching

program Γwith at most O(log |c|)vertices whose path value is c.

Proof. We can assume without loss of generality that c>0: Given a binary algebraic

branching program Γwith path value c>0 and at most O(log c)vertices, we can

add a single vertex t′and an edge from tto t′with label 1 to obtain a new program

(Γ′,s,t′)with path value −c.

For a natural number c, an addition chain of length ℓis a sequence of distinct natural

numbers 1 =c0,c1, . . . , cℓ=ctogether with a sequence of tuples (j1,k1), . . . , (jℓ,kℓ)

such that ci=cji+ckiand ji,ki<ifor all 1 ≤i≤ℓ. However, we will think of

this data as a digraph ˜

Γon the vertices {v0, . . . , vℓ}with edges (vji,vi)and (vki,vi)

for all 1 ≤i≤ℓ. The labels of all edges are equal to 1. Note that we allow double

edges in these digraphs temporarily. We set s:=v0and t:=vℓ. Thus, we view an

addition chain as an acyclic digraph where every vertex except for v0has indegree

two. This already strongly resembles a binary algebraic branching program, but ˜

might have parallel edges. Observe that there are exactly cimany paths from v0to vi

in the digraph ˜

Γ. In particular, there are exactly cpaths from sto tin ˜

Γ.

Using the algorithm of repeated squaring [Knu98, Sec. 4.6.3, eq. (10)] one can

construct an addition chain ˜

Γas above with at most O(log c)vertices and such that

there are exactly cpaths from sto tin ˜

Γ. For every edge (v,w)in ˜

Γwe add a new

vertex uand replace the edge (v,w)by two new edges (v,u)and (u,w). We call the

resulting digraph Γ= (Γ,s,t). Observe that the binary algebraic branching program Γ

has no parallel edges any more and all s-t-paths in Γhave even length. Also, the

digraph Γstill has O(log c)many vertices. Labelling all edges in Γwith 1, the path

value of Γis equal to c.

2.1.2 Proposition. Let Cbe a d×dmatrix whose entries are variables and arbitrary

integer entries. Let cmax be the integer entry of Cwith the largest absolute value. Then

there is a binary variable matrix Aof size O(d2·log |cmax|)with det(A) = det(C).

Proof. We will interpret Cas the directed adjacency matrix of a digraph. Any edge

that has an integer label which is neither 1 nor 0 will be replaced by a subgraph

of size O(log |cmax|)arising from the construction of the previous Proposition 2.1.1.

The directed adjacency matrix of the resulting graph will be the desired matrix A.

Formally, we proceed by induction.

Denote by kthe number of integer entries in the matrix Cthat are neither equal

to 0 nor 1. By induction on k, we will prove the slightly stronger statement that there

x3y

C=⎛

⎜

⎝

3 0 2

0x0

x0y⎞

⎟

⎠

det(C) = 3xy −2x2

Figure 2.1.1: Given a matrix Cwe construct a digraph Hwith directed adjacency matrix C(left hand

side) and the digraph G(right hand side) by replacing the edge with label 2 in Hby a binary algebraic

branching program. We omit the labels for edges that have label 1. The right hand side depicts the

cycle covers Kof Gand the left hand side shows the corresponding cycle covers KHof H.

is a binary variable matrix Aof size d+k·O(log |cmax|)with det(A) = det(C). Since

k≤d2, this implies the statement. Note that the case k=0 is trivial, so we assume

k≥1 and perform the induction step.

Let Hbe the digraph whose directed adjacency matrix is C. Recall that this means

the following: His a digraph on the vertices 1, . . ., dand there is an edge (i,j)with

label Cij if Cij =0 and otherwise no such edge exists. Let e= (i,j)be the edge

corresponding to an integer entry c=Cij which is neither 0 nor 1. Let Γ= (Γ,s,t)be

a binary algebraic branching program with path value cand O(log |c|)many vertices,

which exists by Proposition 2.1.1.

We will now replace the edge (i,j)by Γ(see Figure 2.1.1): Let Gbe the digraph that

arises from H∪Γby removing the edge (i,j), adding edges (i,s)and (t,j)with label

1 and adding loops with label 1 to all vertices of Γ. The directed adjacency matrix of

Ghas size d+O(log |c|)≤d+O(log |cmax|)and contains k−1 integer entries which

are neither 0 nor 1. By applying the induction hypothesis to the directed adjacency

matrix of G, we obtain a matrix Aof size

whose determinant equals the value of G. We are left to show that the value of Gis

equal to det(C), i.e., the value of H.

For this purpose, we will analyze the relation between cycle covers of Gand H,

which is straightforward (see Figure 2.1.1): Consider a cycle cover Kof G. Any vertex

of Γwhich is not covered by its loop must be part of a cycle whose intersection with Γ

is a path from sto t. To Kwe can therefore associate a cycle cover KHof Has follows:

If every vertex of Γis covered by its loop in K, let KHbe Kwithout these loops.

Otherwise, there is a unique cycle κKin Kthat restricts to an s-t-path πKin Γ. Let κH

be the intersection κK∩Htogether with the edge (i,j)and note that κH

Kis a cycle in H.

We obtain KHfrom Kby replacing κKwith κH

Kand removing all remaining loops from

inside Γ.

All cycle covers Lof Hare of the form L=KHfor some cycle cover Kof G. If Lis

a cycle cover of Hcontaining the edge (i,j)then the cycle covers Kof Gwith L=KH

are in bijection with the s-t-paths in Γ. We now fix such a cycle cover L. By definition

of the value of a digraph, it suffices to show that

∑

Kcycle cover of G

such that L=KH

wt(K) = wt(L).

Note that Kand L=KHdiffer only in loops and in the cycles κKand κH

K, respectively.

Since loops contribute a factor of 1 to the weight of a cycle cover, we are left to prove

that

∑

Kcycle cover of G

such that L=KH

wt(κK) = wt(κH

K).

Let e1, . . . , erbe the labels of the edges of κK∩H. These are the edges shared by κK

and κH

K. Thus,

wt(κH

K) = (−1)r·c·e1···er=⎛

⎜

⎝∑

πis s-t-path

inside P

wt(π)⎞

⎟

⎠·(−1)r·e1···er

=⎛

⎜

⎝∑

Kcycle cover of G

such that L=KH

wt(πK)⎞

⎟

⎠·(−1)r·e1···er

=⎛

⎜

⎝∑

Kcycle cover of G

such that L=KH

wt(πK)·(−1)r·e1···er⎞

⎟

⎠=∑

Kcycle cover of G

such that L=KH

wt(κK)

is precisely the desired equality.

2.1.3 Corollary. For every polynomial P∈Z[x]there exists a binary variable matrix

whose determinant is P.

Proof. Combine Theorem 1.5.4 and Proposition 2.1.2.

2.2 Lower Bounds

This section is dedicated to the proof of Theorem 2.0.2. Let B:={0,1}. A sequential

numbering makes the proof much easier to read, so we think of the variables as

arranged in a 3 ×3 matrix

x=⎛

⎜

⎝

x1x2x3

x4x5x6

x7x8x9

⎞

⎟

⎠.

In this section, we will understand per3=per(x)as a polynomial in the variables

x1, . . . , x9instead of the variables xij with 1 ≤i,j≤3.

2.2.1 Proof Outline

Let d∈Nand Aan d×dbinary variable matrix. The binary matrix B(A)∈Bd×d

is defined as the matrix arising from Aby setting all variables to 1. We call B(A)the

support matrix of A. If we set all variables to 1 in per3, we obtain the value 6, so if

per3=det(A), then substituting 1 for all variables on both sides of the equation, we

obtain the condition

6=det(B(A)). (1)

In [EZ62;Slo11], the maximal values of determinants of binary matrices are computed

for small values of d. Since

∀B∈B5×5: det(B)≤5, (2)

we immediately obtain the lower bound bdc(per3)≥6.

Unfortunately, there are several matrices B∈B6×6that satisfy det(B) = 6. We

proceed in two steps to verify that nevertheless, none of these matrices Bis the support

matrix B(A)of a candidate matrix Awith per3=det(A). A rough outline is the

following:

(a) Enumerate all matrices B∈B6×6with det(B) = 6 up to symmetries.

(b) For all those matrices Bprove that Bis not the support matrix B(A)of a binary

variable matrix Awith det(A) = per3. We describe this process in the next

subsection.

2.2.2 Stepwise Reconstruction

Let us make (b) precise. In the hope of failing, we attempt to reconstruct a binary

variable matrix Athat has support Band which also satisfies det(A) = per3. During

the reconstruction process, we successively replace 1’s in Bby the next variable. The

process is as follows:

Given a binary matrix B∈B6×6, let

S:={(i,j)⏐⏐Bij =1}

be the set of possible variable positions. For any set of positions I⊆S, we consider

the matrix BIthat arises from Bby placing a variable yin every position in I. If Bis

the support of a binary variable matrix Awith det(A) = per3and Icontains exactly

the positions where y:=x1occurs in A, then det(BI)must be equal to

per3⎛

⎜

⎝

y1 1

1 1 1

1 1 1⎞

⎟

⎠=2y+4. (3)

We define the set

S:={I⊆S|det(BI) = 2y+4}.

2.2.1 Lemma. Let Abe a binary variable matrix with support Band det(A) = per3.

Let k∈{1, . . . , 9}and define Ik:={(i,j)⏐⏐Aij =xk}as the set of positions where the

variable xkoccurs in A. Then, we have Ik∈ S.

Proof. By the symmetry of the permanent, we may assume that k=1. In the matrix A,

setting every variable except y:=x1to 1 yields the matrix BIand therefore, det(BI) =

2y+4 as in (3), because det(A) = per3. This means Ik∈ S by definition.

Therefore, if Bis the support matrix B(A)of a binary variable matrix Awith

det(A) = per3, we can find 9 pairwise disjoint sets in S, one for each variable xk,

that specify precisely where to place these variables in A. By a recursive search and

backtracking, we now look for sets I1, . . . , Ik∈ S such that

iI1, . . . , Ikare pairwise disjoint.

ii Placing xiinto Bat every position from Iifor 1 ≤i≤kyields a matrix Aksuch

that det(Ak)is equal to per3(x1, . . . , xk, 1, . . . , 1).

The search is recursive in the following sense: First, the possible choices at depth k=1

are given by S. Enumerating the possible choices for depth k+1 works as follows: For

each choice I1, . . . , Ik∈ S with the above two properties, we enumerate all Ik+1∈ S

that have empty intersection with I1∪··· ∪ Ikand check whether condition (ii) is

satisfied.

If the recursive search never reaches k=9 or fails there, then Bis not the support

of a binary variable matrix Awith det(A) = per3. If we reach level 9 however and do

not fail there, we have found such an A.

In practice, the evaluation of det(A)is sped up significantly by working over a

large finite field Fpand choosing random elements x1, . . . , x9∈Fp\{0,1}.

2.2.3 Exploiting Symmetries in Enumeration

Let us call two matrices equivalent if they arise from each other by transposition and/or

permutation of rows and/or columns. A key observation is that equivalent matrices

have the same determinant up to sign. Therefore we do not have to list all binary

matrices B∈B6×6with det(B) = 6, but it suffices to list one representative matrix

Bwith det(B) = ±6 for each equivalence class. It happens to be the case that the

equivalence classes of 6×6 binary matrices are in bijection to graph isomorphy classes

of undirected bipartite graphs G= (V∪W,E)with |V|=|W|=6, V∩W=∅as

follows: For V={v1, . . . , v6}and W={w1, . . . , w6}, the bipartite adjacency matrix

B(G)∈B6×6of Gis defined via B(G)i,j=1 if and only if {vi,wj}∈E. Row and

column permutations in B(G)are reflected by renaming vertices in G. Transposition

of B(G)amounts to switching Vand Win G.

The computer software nauty [MP13] can enumerate all 251610 of these bipar-

tite graphs, which is already a significant improvement over the 236 =68719476736

elements of B6×6. To further limit the number of bipartite graphs that have to be

considered, we make the following observations:

• We need not consider binary matrices Bcontaining a row iwith only a single entry

Bij equal to 1. Indeed, Laplace expansion over the i-th row yields that det(B)is

equal to the determinant of a 5 ×5 binary matrix, which can at most be 5, see (2).

Translating to bipartite graphs, we only need to consider those bipartite graphs

where all vertices have at least two neighbours.

• If two distinct vertices in Ghave the same neighbourhood, then the bipartite adja-

cency matrix B(G)has two identical rows (or columns) which would imply

det(B(G)) = 0. Hence, we only need to enumerate bipartite graphs where all ver-

tices have distinct neighbourhoods. Unfortunately nauty can impose this restriction

only on rows and not on columns.

With these restrictions, the nauty command

genbg -d2:2 -z 6 6

generates 44384 bipartite graphs, only 263 of which have a bipartite adjacency matrix

with determinant equal to ±6. We then preprocess this list by swapping the first two

rows of any matrix with negative determinant.

Finally, the stepwise reconstruction (Subsection 2.2.2) fails for all of these 263 ma-

trices, proving that bdc(per3)≥7. The algorithm takes 28 seconds on an Intel Core™

i7-4500U CPU (2.4 GHz) to finish.

Unfortunately, bdc(per4)can currently not be determined in this fashion because

the enumeration of all apropriate bipartite graphs, already on 9 +9 vertices, is infea-

sible.

2.3 Uniqueness of Grenet’s construction in the 7×7case

The methods from Section 2.2 can be used to determine all 7 ×7 binary variable

matrices Awith the property that det(A) = per3. By means of a cluster computation

over the course of one week, we determined all 463 binary variable matrices with this

property and made some noteworthy discoveries.

The Grenet construction (see Section 2.5) yields the matrix

⎛

⎜

⎝

x11 x12 x13 0000

100x32 x33 0 0

010x31 0x33 0

0010x31 x32 0

000100x23

000010x22

000001x21

⎞

⎟

⎠

. (4)

It is the unique “sparse” 7×7 binary variable matrix from among the 463, in the sense

that every other matrix from the list has more than three nonzero entries in some row

or column.

Motivated by the above observation, we verified by hand (with computer support)

that in fact, all of the 463 matrices can be reduced to (4) by means of elementary row

and column operations. This can be summarized as follows:

2.3.1 Proposition. Every 7 ×7 binary variable matrix Awith det(A) = per3is equiv-

alent to the Grenet construction (4) under the following two group actions:

(1) The action of{(g,h)|det(g) = det(h)}⊆GL7(Z)×GL7(Z)on 7 ×7 matrices via

left and right multiplication, together with transposition of 7 ×7 matrices.

(2) The action of S3×S3on the variables xij with 1 ≤i,j≤3, and the corresponding

transposition (i.e., the map xij ↦→ xji.)

Note that (1) leaves the determinant of any 7 ×7 binary variable matrix invariant and

(2) leaves the permanent polynomial invariant.

2.3.2 Example. One of the matrices that occur in our enumeration is the matrix

A:=

⎛

⎜

⎝

x31 x32 x31 0x32 1x23

1x33 0x31 x33 x31 x22

x33 0x33 x32 1x32 x21

101000x22

0x11 x12 x13 000

010010x21

000101x23

⎞

⎟

⎠

One can check that indeed det(A) = per3. In this case, the matrices

g:=

⎛

⎜

⎝

0 0 0 0 −1 0 0

0 0 1 0 0 −1 0

0 1 0 −1 0 0 0

1 0 0 0 0 0 −1

0 0 0 0 0 0 1

0 0 0 1 0 0 0

0 0 0 0 0 1 0

⎞

⎟

⎠

,h:=

⎛

⎜

⎝

0 1 0 0 1 0 0

−1 0 0 0 0 0 0

0−1 0 0 0 0 0

0 0 −10000

1 0 0 0 0 1 0

0 0 1 1 0 0 0

0 0 0 0 0 0 1

⎞

⎟

⎠

are both invertible over Zand gAh is precisely (4).

2.4 Algebraic Complexity Classes

In this section we relate binary determinantal complexity to classical complexity mea-

sures. An algebraic circuit is called skew if for every multiplication gate at least one

of its two parents is an input gate. We denote by ccs(P)the minimum number dsuch

that Pcan be computed by a skew circuit of size d.VPsconsists of all p-families P

such that ccs(Pm)∈poly(m).

If all input gates of a circuit Care labeled with either 1, −1, or a variable, the circuit

is called constant-free. Note that every constant-free circuit computes a polynomial

that has integer coefficients. We denote by cc0

sand cc0

ws analogously defined complex-

ity measures with the additional condition that only constant-free circuits are allowed.

The complexity classes VP0

sand VP0

ws are defined as consisting of all p-families Psuch

that cc0

s(Pm)and cc0

ws(Pm)are respectively sequences in poly(m). See also [Mal03].

A fundamental result in [Tod92] (see also [MP08]) is that VPws =VPs, so VPsis

another class for which the determinant is complete – recall Subsection 1.5.1. Analyz-

ing the constants which appear in the proof of VPws =VPsin [Tod92], we see that the

proof immediately yields VP0

ws =VP0

s. For the sake of comparison with VP0

s, let us

make the following definition.

2.4.1 Definition. The complexity class DETP0consists of all sequences of polynomials

that have polynomially bounded binary determinantal complexity bdc.

The main purpose of this section is to show the following statement.

2.4.2 Proposition. VP0

ws =VP0

s=DETP0.

Proof. The proof of [Tod92, Lemma 3.4] immediately shows that DETP0⊆VP0

s. To

show that VP0

s⊆DETP0we adapt the proof of [Tod92, Lemma 3.5 or Theorem 4.3],

but a subtlety arises: The proof shows that from a weakly skew or skew circuit Cwe

can construct a matrix A′of size polynomially bounded in the number of vertices in C

such that det(A′)is the polynomial computed by Cwith the drawback that A′is not

a binary variable matrix, but A′has as entries variables and constants 0, 1, and −1.

Fortunately Proposition 2.1.2 establishes DETP0=VP0

s=VP0

ws.

2.4.3 Remark. In the past, other models of computation with bounded coefficients

have already given way to stronger lower bounds than their corresponding unre-

stricted models: [Mor73] on the Fast Fourier Transform, [Raz03] on matrix multipli-

cation, and [BL04] on arithmetic operations on polynomials.

From Valiant’s completeness result [Val79a] we deduce that VP =VNP implies

perm/∈VP0

ws. A main goal is to prove perm/∈VP0

ws unconditionally. This could be a

simpler question than VP =VNP or even VP0=VNP0, because with what is known

today, from perm∈VP0

ws we cannot conclude VP0=VNP0, see [Koi04, Thm. 4.3]. If

we replace the permanent polynomial by the Hamiltonian Cycle polynomial

HCm:=∑

π∈Sm

πis m-cycle

∏

i=1

xi,π(i),

then the question HCm/∈VPws is indeed equivalent to separating VP0

ws from VNP0,

see [Koi04, Thm. 2.5], mutatis mutandis.

We ran our analysis for HCm,m≤4 and proved bdc(HC1) = 1, bdc(HC2) = 2,

bdc(HC3) = 3, bdc(HC4)≥7. The matrices are given at the end of Section 2.5

in (5). This means that 7 ≤bdc(HC4)≤13, where the upper bound follows from

considerations analogous to Grenet’s construction, see Section 2.5.

2.5 Graph Constructions for Polynomials

In this section, we review the proof of Theorem 2.0.1 from [Gre11]. Furthermore,

we use the same methods to prove the following result about the Hamiltonian Cycle

polynomial:

2.5.1 Theorem. We have bdc(HCm+1)≤m·2m−1+1 for all m∈N.

Recall that we denote by [m]:={1, . . . , m}the set of numbers between 1 and m.

2.5.1 Grenet’s Construction for the Permanent

We prove Theorem 2.0.1. The construction of Grenet is a digraph Γwhose vertices

V:={vI|I⊆[m]}are indexed by the subsets of [m]. Hence, d:=|V|=2m. We

partition V=V0∪···∪Vmsuch that Vicontains the vertices belonging to subsets

of size i. We set s:=v∅and t:=v[m], so V0={s}and Vm={t}. Edges will go

exclusively from Vi−1to Vifor i∈[m]. In fact, we insert an edge from vIto vJif

and only if there is some j∈[m]with J=I∪{j}. This edge is then labeled with the

variable xij, where i=|J|. For example, there are medges going from V0to V1, one

for each variable x1jwith 1 ≤j≤m. It is clear that for each permutation π∈Sm,

there is precisely one s-t-path in Γwhose path weight is (−1)m−1·x1,π(1)···xm,π(m).

Consequently, the path value of the algebraic branching program Γ= (Γ,s,t)is equal

to (−1)m−1·perm. Theorem 2.0.1 then follows from the following lemma:

2.5.2 Lemma. Let Γ= (Γ,s,t)be a binary algebraic branching program on d≥3

vertices with path value ±P. Then, there is a binary variable matrix of size d−1

whose determinant is equal to P.

Remark. The proof of this lemma is essentially identical to the proof of Theorem 1.5.4,

but it seems convenient to present it anyway.

Proof. We first construct a graph Gfrom Γby identifying the two vertices sand t

and adding loops with label 1 to every other vertex. The s-t-paths in Γare then in

one-to-one correspondence with the cycle covers of G: Indeed, any cycle cover in G

must cover the vertex s=tand this cycle corresponds to an s-t-path in Γ. Every

other vertex can only be covered by its loop because Γis acyclic. The graph Gnow

has the value ±Pby definition and its directed adjacency matrix Ahas size d−1.

Since d−1≥2, we can exchange the first two rows of Ato change the sign of its

determinant.

2.5.2 Hamiltonian Cycle Polynomial

In this subsection, we prove Theorem 2.5.1 using Lemma 2.5.2. In order to construct

a binary algebraic branching program Γ= (Γ,s,t)with path value HCm+1, we pro-

ceed similar to Grenet’s construction for the permanent. We will refer to cyclic per-

mutations in Sm+1of order m+1 simply as cycles because no cyclic permutations

of lower order will be considered. Observe that the cycles in Sm+1are in bijection

with the permutations in Sm. This can be seen by associating to π∈Smthe cy-

cle σ= (π(1), . . . , π(m),m+1)∈Sm+1. In other words, σmaps m+1 to π(1), it

maps π(1)to π(2)and so on.

In addition to two vertices sand t, our binary algebraic branching program will

have a vertex v(I,i)for every nonempty subset I⊆[m]and i∈I. By our above

Lemma 2.5.2, the resulting binary variable matrix will have a size of

∑

i=1(m

i)·i=m·2m−1+1.

For m=3, this is equal to 3 ·22+1=13.

We construct the edges in Γso that every cycle σ= (a1, . . . , am,m+1)corresponds

to an s-t-path which has v(I,i)as its k-th vertex if and only if I={a1, . . . , ak}and

i=ak. We insert the following edges:

• from sto v({i},i)for each i∈[m]with label xm+1,i

• from v(I,i)to v(I∪{j},j)for each i∈I⊆[m]and j∈[m]\Iwith label xi,j

• from v([m],i)to tfor each i∈[m]with label xi,m+1.

We can again partition the set of vertices as V=V0∪. . . ∪Vm+1where V0={s},

Vm+1={t}and for k∈[m], the set Vkconsists of all vertices v(I,i)with|I|=k. Then,

edges go only from Vkto Vk+1, in particular Γis acyclic. Furthermore, all s-t-paths in Γ

have the same lengths and correspond uniquely to cycles in Sm+1. This concludes

the proof of Theorem 2.5.1.

We know of no better construction for arbitrary m, but for small mwe have

HC2=det (x12 0

0x21)HC3=det ⎛

⎜

⎝

0x12 x13

x21 0x23

x31 x32 0⎞

⎟

⎠. (5)

Chapter 3

Geometric Complexity Theory

Recall Definition 1.3.4 for the notation P≤RQwhen P,Q∈R[x]are polynomials

over a commutative ring R. We call two polynomials Pand Qequivalent if P≤RQ

and Q≤RP. We write P≃RQin this case and P≃Qif there is no ambiguity

concerning the ring R. This notion has been used for p-families already [Bü00, p. 8]

but we require it here for single polynomials only.

It follows directly from the definition that circuit complexity and (binary) deter-

minantal complexity are examples for complexity measures in the following sense:

3.0.1 Definition. Let Rbe a commutative ring. An arithmetic complexity measure

over Ris a function c:R[x]→Nsuch that P≤RQimplies c(P)≤c(Q).

Remark. For every complexity measure c, equivalent polynomials clearly have the

same complexity with respect to c. Observe also that equivalent polynomials are

identical up to renaming the variables, which follows easily from the definition. It is

straightforward to check that “≃” is an equivalence relation.

Notation. If Pand Qare two sets of polynomials, we write P ≃ Q to indicate that

(P/≃)=(Q/≃), i.e., the equivalence classes of polynomials represented by Pand Q

are the same.

We will work exclusively over the field R=Cof complex numbers from here on

in, mainly because it simplifies the representation theory and algebraic geometry that

enters the picture later.

3.1 Orbit and Orbit Closure as Complexity Measure

We will now introduce a new complexity measure which is the first step towards a ge-

ometric interpretation of Conjecture 1.5.3. Let Wd:=Cd×dbe the space of d×dsquare

matrices. It is a complex vector space and End(Wd)denotes the set of all endomor-

phisms a:Wd→Wd. The determinant is a map detd:Wd→C, so for a∈End(Wd),

we can consider the composition detd◦a, which is again a polynomial function: This

is just linear substitution of variables. We then define the determinantal orbit com-

plexity of Pas

doc(P):=min{d∈N|∃a∈End(Wd):P≤detd◦a}, (1)

which is well-defined because doc(P)≤dc(P)for all polynomials P. Our main goal

will be to show that the two measures are actually equivalent in the following sense:

3.1.1 Definition. Let Xbe a set and c1,c2:X→Ntwo functions. We say that c1and

c2are polynomially equivalent if there exist t1,t2∈poly such that c1(x)≤t2(c2(x))

and c2(x)≤t1(c1(x)) for all x∈X. We denote this by c1≡c2.

3.1.2 Proposition. We have dc ≡doc.

Proof. The inequality doc ≤dc is clear. The following Lemma 3.1.3 states that there is

a function t∈poly such that dc(detd◦a)≤t(d)for every a∈End(Cd×d). Since any

polynomial Pwith d:=doc(P)admits a linear transformation a∈End(Cd×d)with

P≤detd◦a, we have dc(P)≤dc(detd◦a)≤t(d) = t(doc(P)).

3.1.3 Lemma. There exists a function t∈poly such that for all d∈Nand all linear

maps a:Cd×d→Cd×d, we have dc(detd◦a)≤t(d).

Proof. There is some s∈poly and weakly skew circuits ˜

Cdcomputing the polynomial

detdwith ⏐⏐˜

Cd⏐⏐≤s(d), see [MV97]. Let now a:Cd×d→Cd×dbe a linear map and

denote by aij :Cd×d→Cits (i,j)-th component. Each aij is a linear form in n=d2

many variables. Note that a linear form ∑n

i=1aixiwith a1, . . . , an∈Ccan be computed

by a weakly skew circuit of size 2n−1:

···

× × ··· ×

a1x1a2x2··· anxn

(2)

We construct a weakly skew circuit Cdcomputing detd◦aas follows: Take the disjoint

union of ˜

Cdand d2circuits of the form (2), each of which respectively computes aij for

all 1 ≤i,j≤d. Then, identify the input gate labeled xij in ˜

Cdwith the gate computing

aij. The result is a weakly skew circuit Cwhich computes detd◦aand|C|<|˜

Cd|+2d4.

Hence Theorem 1.5.2 implies

dc(detd◦a)≤|C|+1≤ | ˜

Cd|+2d4≤s(d) + 2d4=t(d).

3.2 Border Complexity

Let Wd:=Cd×dbe the space of d×dsquare matrices. By Proposition 3.1.2, Conjec-

ture 1.5.3 can be expressed in terms of the values d,m∈Nwhere permis a projection

of some polynomial in detd◦End(Wd).

3.2.1 Remark. Unfortunately, detd◦End(Wd)is unattractive from a geometric point

of view because it is neither a closed nor an open set for the Euclidean or the Zariski

topology in general. We will now replace this set by its closure, which corresponds

to allowing arbitrary approximation in our computational model. This is arguably

natural from both a computational and a geometric point of view, but the implications

for the computational model are not completely understood yet.

For y={y1, . . . , yn}⊆xand d∈N, the space C[y]≤d={P∈C[y]|deg(P)≤d}is

a finite-dimensional vector space which we endow with the Euclidean topology and

we consider on C[y]the final topology with respect to the inclusions C[y]≤d⊆C[y].

Lastly, the topology on C[x]that we use is the final topology with respect to the

inclusions C[y]⊆C[x]for all finite subsets y⊆x.

For any complexity measure c:C[x]→N, we define the corresponding border

complexity function as

c(P):=min{d∈N⏐⏐⏐P∈c−1([d])},

where we use the notation [d]:={1, . . . , d}and hence, c−1([d]) = {P|c(P)≤d}. If c

measures complexity, then cmeasures approximate complexity: c(P)≤dmeans that

any neighbourhood of Pcontains a polynomial of complexity at most d. Clearly,

c(P)≤c(P)always holds.

This defines the determinantal orbit border complexity doc:C[x]→Nand the

determinantal border complexity dc:C[x]→N. We note that dc and doc are poly-

nomially equivalent by general principle:

3.2.2 Proposition. For two complexity measures c1and c2,c1≡c2implies c1≡c2.

Proof. There is a t∈poly such that c1(P)≤t(c2(P)) for all P∈C[x]. In other

words, we have c−1

2([d]) ⊆c−1

1([t(d)]). If P∈C[x]satisfies d:=c2(P), then it follows

from an elementary topological argument that P∈c−1

2([d]) ⊆c−1

1([t(d)]), hence

c1(P)≤t(d) = t(c2(P)). The statement follows by symmetry.

With Proposition 3.1.2 we obtain:

3.2.3 Corollary. doc ≡dc.

The precise statement of the conjecture by Mulmuley and Sohoni is the following:

3.2.4 Conjecture ([MS01, 4.3]). doc(perm)is not polynomially bounded in m.

While the converse is not known, Conjecture 3.2.4 implies Conjecture 1.5.3 be-

cause we have doc(perm)≤doc(perm): If doc(perm)/∈poly(m), then we also have

doc(perm)/∈poly(m). For homogeneous polynomials such as the permanent, we can

give a more concrete description of doc. We denote by GL(Wd)⊆End(Wd)the set of

invertible endomorphisms, the general linear group on Wd.

3.2.5 Proposition. For a homogeneous polynomial P∈C[x]mand any x∈xwith

x/∈supp(P), we have

doc(P) = min{d∈N⏐⏐⏐∃Q∈detd◦GL(Wd):xd−mP≃Q}.

For the proof, we require the following observations:

3.2.6 Lemma. We have detd◦End(Wd) = detd◦GL(Wd).

Proof. We only have to show the inclusion “⊆”. Since the right hand side is closed, it

is in fact sufficient to show that it contains detd◦End(Wd). Consider the map

ω: End(Wd)−→ C[x]

a↦−→ detd◦a

It is continuous because the coefficients of detd◦aare polynomials in the entries of (a

matrix representation of) a. Since GL(Wd)is dense in End(Wd), we have

detd◦End(Wd) = ω(End(Wd)) = ω(GL(Wd)) ⊆ω(GL(Wd)) = detd◦GL(Wd).

3.2.7 Lemma. For two sets P,Q ⊆ C[x]of polynomials, P ≃ Q implies P ≃ Q.

Proof. Since “≃” is an equivalence relation, we can give C[x]/≃the quotient topology

and we claim that the quotient map π:C[x]→C[x]/≃closed. Let y⊆xbe a finite

subset. We denote by Sythe group of all set automorphisms of y. For P,Q∈C[y],

we then have P≃Qif and only if there is some π∈Sywith P=Qπ, in the sense

of Definition 1.3.4. Hence, the equivalence class of any P∈C[y]is finite, from which

it follows easily that the quotient map C[y]→C[y]/≃is closed. Since C[x]/≃has

the final topology with respect to πand C[x]has the final topology with respect to

the inclusions C[y]⊆C[x]for any finite y⊆x, it follows that πis closed. Therefore,

π(P) = π(Q)implies π(P) = π(P) = π(Q) = π(Q).

Proof of Proposition 3.2.5.Let x∈xand ˜

x:=x\{x}. Note that every polynomial is

equivalent to a polynomial in C[˜

x]. The map hd:C[˜

x]m→C[x]dgiven by P↦→ xd−mP

is linear, in particular it is continuous.

Claim. If doc(P)≤d, then there is some a∈End(Wd)with hd(P)≃detd◦a.

We will prove this claim later and show first that it implies the statement. Let docm

be the restriction of doc to C[x]mand observe

doc−1

m([d]) ={P∈C[x]m|doc(P)≤d}

={P∈C[x]m|∃a∈End(Wd):P≤detd◦a}(by the claim)

≃{P∈C[˜

x]m|∃a∈End(Wd):hd(P) = detd◦a}(by Lemma 3.2.7)

=h−1

d(detd◦End(Wd)) (by definition)

=h−1

d(detd◦End(Wd)) (hdis continuous)

=h−1

d(detd◦GL(Wd)).(by Lemma 3.2.6)

We are left to prove our claim. Assume that doc(P)≤d, so there is an a∈End(Wd)

with P≤detd◦a. Potentially replacing Pby an equivalent polynomial, we may

assume that supp(P)⊆supp(detd). Let S⊆supp(detd)and σ:S→S∪Cbe the map

with P= (detd◦a)σ. After composing awith an appropriate linear transformation,

we can assume that σ:S→C×, since mapping a variable to another variable or to

zero is a linear operation on the parameter space. Since scaling the variables by a

nonzero complex number is also a linear operation, we achieve that σ:S→{1}. If

S=∅, then P=detd◦aand in particular m=deg(P) = dso we are done. Otherwise,

we can compose awith a linear projection mapping all variables in Sto one x∈Sand

consequently replace Sby{x}. Since Pand detd◦aare both homogeneous, it follows

that xd−mP=detd◦a.

3.3 The Flip via Obstructions

In this section, we freely use concepts and results from representation theory and

geometric invariant theory, see Chapter A. We also expect the reader to be familiar

with elementary topology.

In light of Proposition 3.2.5 and Conjecture 3.2.4, for m≤d, we define the padded

permanent ppd,m:=xd−m

d,d·permwhich is a degree dform on the space W=Cd×d.

3.3.1 Remark. The padding arises naturally from a computational perspective, but it

causes significant problems from a geometric point of view. We will point out these

problems when they occur and conclude the discussion in Chapter 4.

3.3.2 Definition. For P∈C[W]d, we define Ω(P):=P◦GL(W)the orbit of P. We

denote by Ω(P)the Euclidean closure of Ω(P). We also write ΩPand ΩPinstead

of Ω(P)and Ω(P), depending on which is more readable.

Let Wbe a finite-dimensional complex vector space and P,Q∈C[W]dtwo homo-

geneous forms of degree don W. In the context of Conjecture 3.2.4, they play the roles

of Q=ppd,mand P=detdon W=Cd×d, for certain values of d,m∈N.

By Proposition 3.2.5, we are interested in a way to prove that Q/∈ΩP. By The-

orem A.1.9.(1), the set ΩPis locally closed and the following classical result implies

that ΩPis also Zariski closed, hence an affine variety.

3.3.3 Theorem ([Kra85, AI.7.2]). Let Xbe a complex variety. For a constructible set

U⊆X, the closure of Uin the Euclidean and in the Zariski topology coincide.

It is the goal to show that Q/∈ΩP. Assuming the converse, we get

Q∈P◦GL(W)⇐⇒ Q◦GL(W)⊆P◦GL(W)⇐⇒ Q◦GL(W)⊆P◦GL(W).

This would imply that there is a surjection π:C[ΩP]C[ΩQ]of the corresponding

coordinate rings. Via the induced action of GL(W)on these rings (see Remark A.1.5),

πis also a morphism of GL(W)-modules. Choose a basis W∼

=Cnand by Para-

graph A.2.8, we can write

C[ΩP] = ⨁

λ∈Λ+

V(λ)ocP(λ).

We call ocP(λ)the orbit closure coefficient of Pat λ. Since πis a GLn-equivariant

morphism, Schur’s Lemma (Lemma A.2.2) implies that ocQ(λ)≤ocP(λ)for all λ∈

Λ+

d. Hence:

3.3.4 Proposition (The Flip). Let P,Q∈C[x1, . . . , xn]d. If there exists a λ∈Λ+

nwith

ocQ(λ)>ocP(λ), then Q/∈ΩP.

This observation is called the “flip” because it allows us to prove the non-existence

of efficient algorithms by proving the existence of certain objects.

3.3.5 Remark. It is quite unlikely that Proposition 3.3.4 is sufficient to separate Q

from ΩPfor general Pand Q: The GLn-module structure of the coordinate ring of an

affine variety does not define the variety uniquely, see Example 3.3.6 below. It was

kindly communicated to the author by Michel Brion and further related developments

can be found in [Bri11].

3.3.6 Example. Consider G=SL2acting on the space V=C[x,y]2of quadratic forms

in two variables by precomposition. Let C[V] = C[a,b,c]where we write a quadratic

form as a·x2+b·2xy +c·y2. The discriminant of quadratic forms gives a G-invariant

regular function ∆:=b2−ac ∈C[V]. For any t∈C, the polynomial ∆t:=∆−tis

also G-invariant and the fibers Vt:={Q∈V|∆(Q) = t}=Z(∆t)⊆Vare closed

G-orbits in V, see [Kra85, II.3.3, Beispiel 1, p. 102]. The variety Vtis singular if and

only if t=0, so for example V1is not isomorphic to V0. However, we claim that the

G-module structure of the coordinate ring does not depend on t. Indeed, let I⊆C[V]

be the ideal generated by ∆t. Since ∆tis G-invariant of degree 2, we have C[V]d−2∼

=Id

as G-modules for all d∈Z, where the isomorphism is given by P↦→ P·∆t. Hence,

the G-module structure of Idoes not depend on the choice of tand in particular, the

resulting module structure of C[Vt] = C[V]/Iis independent of t.

3.4 Orbit and Orbit Closure

For this chapter, let Wbe a C-vector space of finite dimension with some choice of a

basis W∼

=Cn. Usually, n=d2and Wis a space of d×dmatrices. We write GL(W)

when coordinates are not required and GLnotherwise.

We study the variety ΩP=P◦GL(W)for some homogeneous form P∈C[W]d.

This is the natural first step towards understanding its closure. By Theorem A.1.9,

we are interested in the stabilizer group GP:={g∈GL(W)|P◦g=P}of P. The

following two results justify that for the rest of this section, we assume GPto be

reductive and consequently, ΩPis an affine variety with coordinate ring C[GL(W)]GP.

For a matrix A∈Cd×d, we denote by Atits transpose.

3.4.1 Theorem ([Fro97]). Let W=Cd×dand let detd∈C[W]dbe the determinant. The

stabilizer group Gdetdis reductive of dimension 2(d2−1). Moreover,

(1) The identity component of the stabilizer of detdis the group

G◦

detd={a∈GL(W)|∃S,T∈SLd:∀B∈W:a(B) = SBT}.

(2) Denoting by t∈GL(W)the map t(B):=Bt, the group Gdetdconsists of the two

connected components G◦

detdand t◦G◦

detd.

3.4.2 Theorem ([Bot67;MM62]). Let W=Cm×mwith m≥3 and let perm∈C[W]m

be the permanent. The stabilizer group Gpermis reductive of dimension 2(m−1).

Moreover,

(1) Denoting by ∆m⊆SLmthe subgroup of diagonal matrices where the product of

all entries on the diagonal equals 1,

G◦

perm={a∈GL(W)|∃S,T∈∆m:∀B∈W:a(B) = SBT}.

(2) We embed Sm⊆GLmas the subgroup of permutation matrices. For σ,τ∈Sm,

let cτ

σ∈GL(W)be the map cτ

σ(B) = σ◦B◦τand let Gσ,τ

perm:=cτ

σ◦G◦

perm.

Denoting by t∈GL(W)the map t(B):=Bt, the group Gpermconsists of the 2m!2

connected components tk◦Gσ,τ

permfor k∈{0,1}and σ,τ∈Sm.

C[ΩP]is also a G-module and the orbit coefficients ocP(λ)are defined by

C[ΩP] = ⨁

λ∈Λ+

V(λ)ocP(λ).

Since ΩPis an open, affine, GLn-invariant subset of ΩP, the restriction morphism

C[ΩP]→C[ΩP]is an injective morphism of GLn-modules. By Schur’s Lemma

(Lemma A.2.2), we therefore have ocP(λ)≤ocP(λ)for all λ∈Λ+

n. While the or-

bit closure coefficients are quite elusive, the orbit coefficients admit a nice description

when GPis reductive:

3.4.3 Proposition. For P∈C[x1, . . . , xn]with reductive stabilizer GPand λ∈Λ+

n, we

have ocP(λ) = dim(V(λ)GP).

Proof. Recall from Theorems A.1.9 and A.2.7 that

C[ΩP] = C[GLn]GP=⨁

λ∈Λ+

V(λ)∗⊗V(λ)GP.

By Proposition A.2.10 and because λ↦→ λ∗defines an involution on Λ+

n, we get

C[ΩP] = ⨁

λ∈Λ+

V(λ)⊗(V(λ)∗)GP.

The statement follows by applying Proposition A.2.12 to the representation V(λ)∗of

the reductive group GP.

The orbit coefficients of the determinant and the padded permanent have been

analyzed in [Bür+11].

3.4.1 Characterization by the Stabilizer

It is crucial to determine whether determinant and padded permanent suffer from the

problem raised in Remark 3.3.5. The following special case of a result due to Larsen

and Pink is interesting in this context. We implicitly use Proposition 3.4.3.

3.4.4 Theorem ([LP90]). Let P∈C[x1, . . . , xn]dbe such that GPis semisimple and

connected. If ϱ:GP↪→GLnis an irreducible GP-representation, then GPand ϱare

uniquely determined by the orbit coefficients ocP:Λ+

n→N, up to an isomorphism of

groups and representations, respectively.

The theorem is interesting because it implies that for certain polynomials orbit

coefficients are sufficient to separate orbits. Mulmuley and Sohoni have shown that

the determinant and the permanent are characterized by their stabilizer [MS01] in the

following sense:

3.4.5 Definition. Let Gbe an algebraic group acting linearly on a C-vector space W.

A point P∈Wis said to be characterized by its stabilizer if WGP=C·P.

With this notion, we have:

3.4.6 Corollary. Assume that Pand Qsatisfy the assumptions of Theorem 3.4.4, we

have Q/∈ΩPand Pis characterized by its stabilizer. Then, there exists a λ∈Λ+

nwith

ocQ(λ)=ocP(λ).

Proof. We assume that Pand Qhave the same orbit coefficients and deduce Q∈ΩP.

By Theorem 3.4.4,GP=g−1·GQ·g=GQ◦gfor some g∈GLn, this is the definition

of the two representations being isomorphic. Hence, Q◦g∈C[x1, . . . , xn]GP

d=C·P

because Pis characterized by its stabilizer. In other words, Q∈ΩP.

Unfortunately, the stabilizer groups of determinant and permanent are both dis-

connected, and only G◦

detdis semisimple, G◦

perm∼

=G2(m−1)

mis a torus, therefore only

reductive. It has been established in [AYY13, Thm. 1.5] that the semisimplicity hy-

pothesis in the general theorem by Larsen and Pink [LP90] cannot be dropped.

In the literature, the map λ↦→ ocP(λ)is referred to as the dimension datum of GP.

One can study the much more general case that Gis an algebraic group and H⊆G

is a closed subgroup. The dimension datum of this inclusion is the map Irr(G)→N

given by V↦→ dim(VH). Proposition 3.4.3 states that this is equivalent to the map

ocP:Λ+

n→Nin our situation. The question of how much information about the

inclusion H⊆Gis encoded in the dimension datum is an active and recent area of

research, for some advances see [Yu16].

It seems that there is no version of Theorem 3.4.4 which is taylored to the situa-

tion we consider here, yet. We note however that the stabilizers of determinant and

permanent act irreducibly on W=Cd×d.

3.4.7 Proposition. The space W=Cd×dis an irreducible G◦

detd-representation and in

particular, an irreducible Gdetd-representation.

Proof. Since W∼

=Cd⊗Cd, it is an irreducible (SLd×SLd)-representation [FH04, Ex-

ercise 2.36]. This is precisely the action of G◦

detd.

3.4.8 Proposition. The space W=Cm×mis an irreducible Gperm-representation.

Proof. Note first that W=∑m

i=1∑m

j=1C·Eij where Eij is the matrix with entry 1 in

position (i,j)and 0 everywhere else. Using notation from Theorem 3.4.2, the group

∆m×∆m∼

=Gm−1

m×Gm−1

mis a torus and C·Eij is a weight space for the corresponding

action by simultaneous left and right multiplication. This is the action of G◦

perm. Let

now U⊆Wbe some nonzero, Gperm-stable subspace of W. Then, Uis in particular

stable under G◦

permand must therefore be a direct sum of certain weight spaces. Hence,

there are indices iand jwith Eij ∈U. By Theorem 3.4.2, arbitrary row and column

permutations are in Gperm, so Eij ∈Ufor all 1≤i,j≤m. Hence, W=U.

Remark. While this sounds good, closer inspection reveals that the padded perma-

nent fails to retain the desired property. This is related to the padding variable, as

announced in Remark 3.3.1.

Let m<dbe natural numbers and Q:=ppd,mthe padded permanent. Then,

the space W=Cd×dis certainly not an irreducible GQ-representation: The one-

dimensional space of matrices with zeros everywhere except in position (d,d)is in-

variant under GQdue to the description of GQin [Bür+11, 5.6].

Remark. Note that W=Cm×mis not an irreducible G◦

perm-representation.

Even if we ignore all these problems for now, it remains an open question whether

the orbit closure and its embedding are uniquely defined by the orbit closure coeffi-

cients. This is the important question of whether or not Proposition 3.3.4 is sufficient:

3.4.9 Question. Let P,Q∈C[x1, . . . , xn]dhave reductive stabilizer and Q/∈ΩP. As-

sume that Pis irreducible, characterized by its stabilizer, and Cnis irreducible as

aGP-module.

(1) Does there exist a λ∈Λ+

nwith ocQ(λ)>ocP(λ)?

(2) The same question, but under the additional assumption that Q=xd−m˜

Qis the

product an irreducible polynomial ˜

Q∈C[x1, . . . , xn]mof degree m<dwith a

power of a linear form x, such that ˜

Qis characterized by its stabilizer and Cnis

irreducible as a G˜

Q-module.

(3) The same question, but even more specifically with P=detdand Q=ppd,m.

Chapter 4

Occurrence Obstructions

The Flip (Proposition 3.3.4) offers a way to prove noncontainment of orbit closures.

For the polynomials P=detdand Q=ppd,mit is proposed in [MS01;MS08] to exhibit

weights λwhich satisfy ocP(λ) = 0 and ocQ(λ)=0. This implies ocQ(λ)>ocP(λ)

and therefore Q/∈ΩPby Proposition 3.3.4. Such weights λare called occurrence

obstructions.

We present in this chapter the results of the previously published [BIH17]. The

paper features a result (Theorem 4.1.2) about the nature of occurrence obstructions

which suggests that they are quite rare. In the subsequent works [IP16;BIP16],

Bürgisser, Ikenmeyer, and Panova strengthened this result considerably, eventually

proving that occurrence obstructions cannot be used to prove Conjecture 3.2.4. Their

method relies only on the fact that we consider a padded polynomial, see also Re-

mark 3.3.1. This is by far the biggest clue to date that the padding variable, while

computationally harmless, causes severe problems for the geometry.

The auxiliary Propositions 4.3.5 and 4.3.8 remain of independent interest: They

provide an unconditional version of a related statement by Kumar, even though the

bounds are not particularly impressive.

In what follows, we will denote by Dd:=Ω(detd)and Pd,m:=Ω(ppd,m)the orbit

closures of determinant and padded permanent. Let also n:=d2. Both Ddand Pd,m

are closed subvarieties of the N-dimensional affine space AN=C[xij |1≤i,j≤d]d

of degree dhomogeneous forms on Cd×d, where we define N:=(n+d−1

d). The group

GLn∼

=GL(Cd×d)acts on ANby precomposition.

4.1 Weight Semigroups

We use notation from Section A.3 in this section. Let Z⊆ANbe any GLn-invariant

closed subvariety. The group GLnacts on the coordinate ring C[Z]of Zvia precom-

position. We are interested in the set of irreducible GLn-representations occurring

in C[Z]and define

Λ+(Z):={λ∈Λ+

n⏐⏐V(λ)∗⊆C[Z]}. (1)

It is known that Λ+(Z)is a finitely generated submonoid of Λ+

n, cf. [Bri87]. We are

mainly interested in the case where Z=ΩPis an orbit closure, in this case it is easy

to see that |λ|:=λ1+···+λn∈dZfor any λ∈Λ+(Z). We furthermore denote by

ℓ(λ):=min{k|λk=0}the length of λ. If Pdepends on ℓvariables only, then it is

known that ℓ(λ)≤ℓfor all λ∈Λ+(Z), cf. [Bür+11, §6.3].

An occurrence obstruction is some λ∈Λ+(Pd,m)\Λ+(Dd). If we want to prove at

least dc(perm)>m2+1 by exhibting an occurrence obstruction, then we may assume

that d≥m2+1. Since any λin Λ+(Pd,m)satisfies|λ|∈dZand ℓ(λ)≤m2+1≤d,

we have to look for such partitions λoutside of Λ+(Dd).

Before stating our main result, we need to introduce the concept of saturation,

whose relevance for geometric complexity was already pointed out in [Mul07], see

also [BOR09].

For the following compare [MS05, §7.3]. Let Sbe a submonoid of a free abelian

group Fand⟨S⟩the group generated by S. We call Ssaturated if

∀λ∈⟨S⟩:∀k∈N:kλ∈S⇒λ∈S.

The saturation Sat(S)of Sis defined as the smallest saturated submonoid of Fcon-

taining S. It can also be characterized as the intersection of⟨S⟩with the rational cone

generated by S. An element in Sat(S)\Sis called a gap of S. The reason for this

naming becomes apparent from a simple example:

4.1.1 Example. Consider S=N2\{(0,1),(1,0)}, which has N2as its saturation. Re-

placing Sby Sat(S)means filling up the gaps (0,1),(1,0). Generally, understanding

monoids is difficult due to the presence of gaps.

We can now state the main theorem of this chapter:

4.1.2 Theorem. The saturation of Λ+(Dd)contains the set

{λ∈Λ+

n⏐⏐ℓ(λ)≤d,|λ| ∈ dZ},

provided that d>2. Hence, occurrence obstructions must be gaps of Λ+(Dd).

4.1.1 Related Work

Our work is closely related to a result by Shrawan Kumar [Kum15]. A latin square of

size dis an d×dmatrix with entries from {1, . . . , d}such that in each row and in each

column each number occurs exactly once. The sign of a latin square is defined as the

product of the signs of all the row and column permutations. Depending on the sign,

we can speak about even and odd latin squares. The Alon-Tarsi conjecture [AT92]

states that the number of even latin squares of size dis different from the number of

odd latin squares of size d, provided dis even. The Alon-Tarsi conjecture is known to

be true if d=p±1 where pis a prime [Dri98;Gly10].

4.1.3 Theorem (Kumar). Let dbe even. If the Alon-Tarsi conjecture for d×dlatin

squares holds, then dλ∈Λ+(Dd)for all partitions λsuch that ℓ(λ)≤d.

The above two theorems complement each other. Theorem 4.1.2 is unconditional

and also provides information about the group generated by Λ+(Dd). Theorem 4.1.3

is conditional, but gives a very tight bound on the stretching factor, which is dif the

Alon-Tarsi conjecture holds. The proofs of both theorems focus on the the d-th Chow

variety (see Section 4.3 for details), but otherwise proceed differently. Our proof also

gives information on the stretching factor in terms of certain degrees related to the

normalization the Chow variety, but we were so far unable to bound it in a reasonable

way, see Proposition 4.3.5.

4.2 Saturations of Weight Semigroups of Varieties

We consider the following situation. G:=GLn(C)is the complex general linear group

and Udenotes its subgroup consisting of the upper triangular matrices. Vis a finite

dimensional C-vector space and a rational G-module such that scalar multiples of

the unit matrix In∈Gact on Vby nontrivial homotheties, i.e., there is a nonzero

d∈Zsuch that P◦tIn=tdPfor t∈C×and P∈V. Further, Zdenotes a G-

invariant, irreducible, locally closed nonempty subset of V. Then Zis closed under

multiplication with scalars in C×by our assumption on the G-action.

We consider the induced action of Gon the coordinate ring C[Z]of Zdefined

as in Remark A.1.5. We will denote this induced action from the left and by a dot,

i.e., (g.f)(P) = f(P◦g−1)for P∈Z,f∈C[Z]and g∈G. As in (1) we define

the monoid Λ+(Z)of representations of the G-variety Z. We shall interpret Λ+(G)

as a subset of Znand denote by Λ(Z):=⟨Λ+(Z)⟩the group generated by Λ+(Z).

Moreover, we denote by coneQ(Λ+(Z)) the rational cone generated by Λ+(Z), that is,

coneQ(Λ+(Z)) :={k−1λ⏐⏐⏐k∈N,λ∈Λ+(Z)}⊆Qn.

It is easy to check that the saturation of Λ+(Z)is obtained as

Sat(Λ+(Z)) = Λ(Z)∩coneQ(Λ+(Z)). (2)

We denote by Frac(R)the field of fractions of an integral ring R. We have an induced

G-action on the field of fractions C(Z):=Frac(C[Z]) and denote by C(Z)Uits subfield

of U-invariants. Recall that a highest weight vector is a U-invariant weight vector. The

following lemma is well known, but we include its proof for completeness.

4.2.1 Lemma. We have Frac(C[Z]U) = C(Z)U. Moreover, for a highest weight vector

f∈C(Z)U, there exist highest weight vectors p,q∈C[Z]Usuch that f=p/q.

Proof. The inclusion Frac(C[Z]U)⊆C(Z)Uis obvious. Now let f∈C(Z)Uand

consider the ideal J:={q∈C[Z]|qf ∈C[Z]}of C[Z]. Since J=0 we have JU=0,

cf. [Hum98, §17.5]. Choose a nonzero q∈JU. Then p:=qf ∈C[Z]Uand f=p/q,

hence f∈Frac(C[Z]U).

If f∈C(Z)Uis a weight vector, we can argue as before, choosing q∈JUas a

highest weight vector. Then p:=qf is a highest weight vector in C[Z]. The assertion

follows.

4.2.2 Remark. If Y=∅is a G-invariant open subset of Z, then Λ+(Z)⊆Λ+(Y)and

Λ(Z) = Λ(Y). This follows immediately from Lemma 4.2.1.

Suppose now that Zis a closed subset of V, hence an affine variety. Then we have

an induced G-action on the normalization N(Z)and the canonical map π: N(Z)→Z

is G-invariant. Indeed, the integral closure Rof C[Z]in C(Z)is G-invariant and π

corresponds to the inclusion C[Z]↪→R. By construction, we can identify C(N(Z))

with C(Z). Note that Λ+(Z)⊆Λ+(N(Z)) since πis surjective.

4.2.3 Proposition. We have Λ(N(Z)) = Λ(Z)and Sat(Λ+(N(Z))) = Sat(Λ+(Z)).

More precisely, assume that C[N(Z)] is generated as a C[Z]-module by relements.

Then for all λ∈Λ+(N(Z)), there is some k<rsuch that (r−k)·λ∈Λ+(Z).

Proof. Let λ∈Λ+(N(Z)) and f∈C[N(Z)]Ube a highest weight vector of weight λ.

Then f∈C(Z)Uand Lemma 4.2.1 shows the existence of highest weight vectors

p,q∈C[Z]U, say with the weights µ,ν∈Λ+(Z), respectively, such that f=p/q.

Therefore λ=µ−ν∈Λ(Z). This shows the equality for the groups.

Due to (2), it suffices to prove that coneQ(Λ+(Z)) = coneQ(Λ+(N(Z))). Suppose

f∈C[N(Z)] is a highest weight vector of weight λ. Since fis integral over C[Z],

there are e∈Nand a0, . . . , ae−1∈C[Z]such that such that

fe+

e−1

∑

i=0

aifi=0. (3)

We assume that the degree eis the smallest possible.

Note that eis at most the size of a generating set of C[N(Z)] as an C[Z]-module,

as follows from the classical theory of integral extensions, see [AM69, Prop. 5.1 and

the proof of Prop. 2.4].

Consider the weight decomposition ai=∑µai,µof ai, where ai,µhas the weight µ.

Then ai,µfihas the weight µ+iλ. Moreover, fehas the weight eλ. Since the component

of weight eλin (3) must vanish, we have

fe+

e−1

∑

i=0

ai,(e−i)λ·fi=0.

As the degree eis the smallest possible, the above is the minimal polynomial of f. Ap-

plying any element u∈Uto the above equation and using u.f=f, we get the identity

fe+∑e−1

i=0(u.ai,(e−i)λ)fi=0. The uniqueness of the minimal polynomial implies that

u.ai,(e−i)λ=ai,(e−i)λfor all i. Hence ai,(e−i)λis a highest weight vector, provided it is

nonzero. Since there exists i<ewith ai,(e−i)λ=0, we see that (e−i)λ∈Λ+(C[Z])

for this particular i. We conclude that λ∈coneQ(C[Z]).

4.2.4 Example. If we consider instead the torus G= (C×)dwe can identify Λ+

with Zd. Let S⊆Zdbe a finitely generated submonoid and consider the finitely gen-

erated subalgebra C[S]:=⨁s∈S(C·xs1

1···xsd

d)of C[x±1

1, . . . , x±1

d]. If we interpret C[S]

as the coordinate ring of an affine variety Z, then we have a G-action on Zand Λ+(Z)

can be identified with S(Zis called a toric variety). It is known that C[Sat(S)] equals

the integral closure of C[S]in Frac(C[S]), cf. [MS05, Prop. 7.25, p. 140]. Thus the

affine variety corresponding to Sat(S)equals the normalization of Z. This illustrates

Proposition 4.2.3 in the special case of toric varieties.

4.3 Proof of Main Results

Write W:=Cn,G:=GL(W)and consider the symmetric power V:=SymnW∗with

is natural G-action. Using the chosen coordinates on W, we obtain an isomorphism

W→W∗that we denote by a↦→ a∗. We can interpret Vas the space C[W]nof

degree nforms. Note that C[V]k∼

=SymkSymnWas G-modules. Let x1, . . . , xn∈W∗

be a basis and consider the universal monomial

mnn:=x1···xn∈V.

Clearly, Ωmnnconsists of all symmetric products a1···anof nlinearly independent

linear forms ai. We define the n-th Chow variety

Cn:=mnn◦End(W) ={a1···an|a1, . . . , an∈W∗}⊆V.

The name comes from the fact that Cnis a special case of a Chow variety, see [GKZ94].

4.3.1 Lemma. We have Cn=Ωmnn.

Proof. The inclusion Cn⊆Ωmnnfollows since GL(W)is dense in End(W). For the

converse inclusion, let Q∈Ωmnnbe nonzero. By Theorem 3.3.3, it is the limit of a

sequence (tk·a1k···ank)k∈N, where tk∈C×and aik ∈W∗are linearly independent

with ∥aik∥=1. By compactness of the unit sphere in W∗we may assume that after

passing to a subsequence, (aik)k∈Nis convergent for all 1 ≤i≤n. Let bi:=limk→∞aik.

Then, (∏n

i=1aik)k∈Nconverges to b1···bn=0. It follows easily that (tk)k∈Nconverges

to some t∈C×and consequently, Q=t·b1···bn∈Cn.

When we identify xiwith the variable xii, the Chow variety Cnis contained in Dn

by mapping xij to 0 for i=j, c.f. [Lan15]. The basic strategy, as in Kumar [Kum15], is

to replace Dnby the considerably simpler Cnand to exhibit elements in the monoid of

representations of the latter. More specifically, we have Λ+(Cn)⊆Λ+(Dn)and hence

Sat(Λ+(Cn)) ⊆Sat(Λ+(Dn)). Our main Theorem 4.1.2 is an immediate consequence

of the following result.

4.3.2 Theorem. We have Sat(Λ+(Cn)) ={λ∈Λ+

n:|λ|∈nZ}, provided n>2.

According to Proposition 4.2.3, for proving this we may replace Cnby its normal-

ization. It is crucial that the latter has an explicit description.

For the following arguments compare [Bri93] and [Lan15]. We will revisit them in

Section 6.2. The symmetric group Snoperates on the group

Tn:={(t1, . . . , tn)∈Cn|t1···tn=1}

by permutation. The corresponding semidirect product Hn:=TnoSnacts on the

space Wn=W×. . . ×Wby scaling and permutation. Note that this action commutes

with the G-action. Consider the product map

ω:Wn=W×. . . ×W−→ Cn

(a1, . . . , an)↦−→ a∗

1···a∗

which is surjective and G-equivariant. Clearly, ωis invariant on H-orbits. Moreover,

the fiber of a nonzero Q∈Cnis an Hn-orbit. This easily follows from the uniqueness

of polynomial factorization.

The group Gcontains the subgroup {t·idW|t∈C×}∼

=C×and this C×-action

induces a natural grading on the coordinate rings C[Wn]and C[Cn]. Since ωis G-

equivariant, the corresponding comorphism ω♯:C[Cn]→C[Wn]is in particular a

homomorphism of graded C-algebras. However, the C×-action on C[Cn]is not the

canonical one because t∈C×acts by multiplication with the scalar tn. A homoge-

neous element of degree kn in C[Cn]is the restriction of a k-form on V.

The categorical quotient Wn//Hnis defined as the affine variety that has as its

coordinate ring the ring of Hn-invariants C[Wn]Hn, which is finitely generated and

graded since Hnis reductive, cf. [Kra85]. The inclusion C[Wn]Hn↪→C[Wn]defines

aG-equivariant, surjective morphism π:Wn→Wn//Hn. Since Wnis normal, the

quotient Wn//Hnis normal as well, see [Dol03, p. 45] for the easy proof. The map ω

factors through a G-equivariant morphism

φ:Wn//Hn→Cn, (4)

due to the universal property of categorical quotients. Moreover, by construction, the

fibers of φover a nonzero Q∈Cnconsist of just one element. The action of Hn

on C[Wn]is linear, therefore it respects the grading. It follows that the comorphism

φ♯:C[Cn]→C[Wn]Hnis again a homomorphism of graded C-algebras.

The following is shown in [Bri93, Prop., p. 351] and we give a proof in slightly

different language in Section 6.3.

4.3.3 Lemma. The morphism φ:Wn//Hn→Cnis the normalization of Cn.

Furthermore,

4.3.4 Lemma. We have

Λ+(Wn//Hn) ={λ∈Λ+

n⏐⏐⏐∃k:|λ|=kn and V

G(λ)⊆SymnSymkCn}.

Proof. We shall decompose the coordinate ring C[Wn//Hn]with respect to the G-

action. We have C[W] = ⨁k∈NSymkW∗and therefore

C[Wn] = C[W]⊗n=⨁

(k1,...,kn)∈Nn

Symk1W∗···⊗SymknW∗.

Taking Tn-invariants yields

C[Wn]Tn=⨁

k∈N

SymkW∗⊗. . . ⊗SymkW∗.

Taking Sn-invariants gives

C[Wn//Hn]∼

=C[Wn]TnoSn=⨁

k∈N

SymnSymkW∗, (5)

and the assertion follows.

Recalling Theorem 4.1.3, we expect the stretching factor to be n. Without relying

on the Alon-Tarsi conjecture however, the following exponential bound is the best we

can currently provide.

4.3.5 Proposition. Λ+(Cn)generates the rational cone {q∈Qn|q1≥ ··· ≥ qn≥0}.

More precisely: Assume n>2. For each partition λwith ℓ(λ)≤nand |λ| ∈ nZ,

there is some number k<nn2−2nsuch that we have 2k·λ∈Λ+(Cn).

Proof. For the first statement, it is sufficient to show that coneQ(Λ+(Cn)) contains

any partition λwith |λ|=nk and ℓ(λ)≤n. According to Proposition 4.2.3, the

semigroups Λ+(Cn)and Λ+(Wn//Hn)generate the same rational cone. So we need

to show that λlies in coneQ(Λ+(Wn//Hn)).

In [BCI11] (see [Ike12] for a simpler proof) it was shown that V

G(2λ)occurs in

SymnSym2k(Cn). Thus Lemma 4.3.4 with Proposition 4.2.3 imply that λlies in the

rational cone generated by Λ+(Wn//Hn).

We will now make the above reference to Proposition 4.2.3 precise. Recall that the

comorphism of φfrom (4) is an integral extension φ♯:C[Cn]↪→C[Wn]Hnof graded C-

algebras. Note that the grading induced by Gon these two rings is such that only

degrees which are multiples of ncontain nonzero elements. We therefore change

the grading such that degree n·kbecomes degree k. This means that the direct

sum (5) is the grading of C[Wn]Hnand the grading of C[Cn]is the grading induced

by the canonical grading induced by the polynomial ring C[V]. Therefore, C[Cn]is

generated by elements of degree one.

By [DK02, Lemma 2.4.7], there is a system of parameters y0, . . . , yr∈C[Cn]1from

among sufficiently generic linear forms. This means that R:=C[y0, . . . , yr]is a poly-

nomial ring in the yiand C[Cn]is a finite R-module. It follows that C[Wn]Hnis integral

over Rand y0, . . . , yris also a homogeneous system of parameters for C[Wn]Hn.

We note that r+1=dim(Wn//Hn) = dim(Wn)−dim(Hn) = n2−n+1, so

r=n2−n. Furthermore, C[Wn]Hnis Cohen-Macaulay [DK02, Thm. 2.5.5] and by

[DK02, Prop. 2.5.3], it follows that C[Wn]Hnis free as an R-module, i.e., C[Wn]Hn∼

=RD

for some D∈N. Since Dis the number of generators of C[Wn]Hnas an R-module,

Dis a (possibly rough) upper bound for the number of generators of C[Wn]Hnas an

C[Cn]-module. The second assertion follows from Proposition 4.2.3 as soon as we

have verified that D<nn2−2n.

The Hilbert polynomial of RDis a polynomial of degree rwhose leading coefficient

is equal to D

r!. Since (5) gives the grading of C[Wn]Hn, the Hilbert polynomial of

C[Wn]Hnis given, for sufficiently large k, by the map

k↦→ ((k+n−1

n−1)+n−1

n),

whose leading coefficient is 1

n!(n−1)!n. Since r=n2−n, we have D=(n2−n)!

n!(n−1)!n. We

apply Stirling’s approximation:

∀n>0: 1 ≤n!

√2π·e−n·nn+1

2≤e1

12n

to the fraction Dand obtain

D=(n2−n)!

n!(n−1)!n≤√2π·e1

12n·(n2−n)n2−n+1

2·en−n2

√2π·nn+1

2·e−n·√2πn·(n−1)(n−1

2)n·e−(n−1)n

=(e

√2π)n

·e1

12n·nn2−n+1

2·(n−1)n2−n+1

nn+1

2·(n−1)n2−n

=(( e

√2π)n

·e1

12n·(n−1)1−n

  

R(n)

·nn2−2n.

It is easy to see that R(n)is monotonically decreasing for n≥2 and takes a value

smaller than 1 for n=3. Hence, for n>2 we have D<nn2−2n.

We shall now determine Λ(Cn). Since Wn//Hnis the normalization of Cn, Propo-

sition 4.2.3 tells us that Λ(Cn) = ⟨Λ+(Wn//Hn)⟩. The latter is described in terms of

plethysms in Lemma 4.3.4.

Recall W=Cnand G=GL(W). We say that λoccurs in SymnSymkWif V

G(λ)

occurs as a submodule in the latter. We will also make use of the convenient notation

k×d:= (d, . . . , d, 0, . . . , 0)∈Znfor a rectangular partition with krows of length d.

4.3.6 Lemma. Let ℓ∈N. If λoccurs in SymnSymkW, then (1×ℓk) + λoccurs in

Symℓ+nSymkW.

Proof. Let WU=Cw, i.e., wis a highest weight vector of Wwith weight 1 ×1. Then

w⊗ℓk∈SymℓSymkWis a highest weight vector of weight (1×ℓk).

Let f∈SymnSymkWbe a highest weight vector of weight λ. Then the product

w⊗ℓkf∈Symℓ+nSymkWis a highest weight vector of weight (1×ℓk) + λ.

4.3.7 Lemma. Let n≥k≥2, d:=k(k−1)/2, and the partition µof size k2be obtained

by appending to 2 ×da column of length k. Further, let λdenote the partition of size

nk obtained by appending to µa row of length (n−k)k. Then the partition λoccurs

in SymnSymkW.

Proof. The GL2-module ΛkSymk−1C2is one-dimensional, since Symk−1C2is of di-

mension k. Hence it contains a nonzero SL2-invariant. In other words, 2 ×doccurs

in ΛkSymk−1C2. The “inheritance principle” states that 2 ×doccurs in ΛkSymk−1Cn

(compare for instance [Ike12, Lemma 4.3.2]).

Cor. 6.4 in [MM15] implies that µoccurs in SymkSymkW. Finally, Lemma 4.3.6

implies the assertion.

4.3.8 Proposition. Λ+(Cn)generates the group{λ∈Zn:|λ|∈nZ}if n>2.

Proof. Using the software [BKT12], we checked that the partition (2,2,0, . . . , 0)occurs

in Sym2Sym2Wand (6, 3,0, . . . , 0)occurs in Sym3Sym3W. Using Lemma 4.3.6, we

can conclude that (2n−2, 2, 0, . . . , 0)occurs in SymnSym2W, and (3n−3, 3,0, . . . , 0)

occurs in SymnSym3Wif n≥3. (We note that (3, 3,0, . . . , 0)does not occur in

Sym2Sym3W; this is the reason for the assumption n>2.) From Lemma 4.3.4 we

conclude that

λ(2):= (n−1, 1,0, . . . , 0) = (3n−3, 3,0, . . . , 0)−(2n−2, 2,0, . . . , 0)

lies in the group Λgenerated by Λ+(Wn//Hn). Clearly, λ(1):= (n, 0, . . . , 0)∈Λ.

For 3 ≤k≤nlet λ(k)∈Λdenote the partition from Lemma 4.3.7. Then we have

ℓ(λ(k)) = kand λ(k)

k=1. This easily implies that λ(1), . . . , λ(n)generate the group

Λ:={λ∈Zn:|λ|∈nZ}. Since Λ⊆˜

Λis obvious, we conclude that Λ=˜

Λ.

Proof of Theorem 4.3.2.Use (2) with Propositions 4.3.5 and 4.3.8.

4.3.9 Remark. The assumption n>2 in Theorem 4.3.2 is necessary. Indeed, we have

C2=Sym2C2, and one can show that Λ+(Sym2C2)generates the group (2Z)2, com-

pare [FH04, §11.2].

Part II

Orbit Closures of Homogeneous Forms

Chapter 5

Preliminaries

Throughout Part II, we will consider the following situation. We work over the field C

of complex numbers and W∼

=Cnis a C-vector space of dimension n, for which

we sometimes assume some choice of coordinates C[W] = C[x1, . . . , xn], to iden-

tify GL(W)with GLn=GLn(C). The group GL(W)acts on the space C[W]d=

SymdW∗of homogeneous forms of degree d>0 on Wby precomposition from the

right:

C[W]d×GL(W)−→ C[W]d

(P,g)↦−→ P◦g

We view V:=C[W]d∼

=ANas an affine space of dimension N=(n+d+1

d). Note

that C[V]is simply a polynomial ring in the coefficients of all homogeneous degree d

polynomials in nvariables. We will study subvarieties of this space in the language

of algebraic geometry, therefore we usually assume the Zariski topology on V.

This scenario is motivated by the GCT approach that we outlined in Chapter 3. In

that context, one is interested only in special cases, primarily W=Cd×dand the point

P=detd. We put this into a slightly more general context here, one reason being

that we have very little understanding of the determinant orbit closure, even for small

values of d. For what little we know, see Chapter 8. To gain a better understanding of

the phenomena that occur, we believe that easier examples should be studied first.

Another reason is also that the generality we present here offers an interesting and

little explored mathematical problem: Classical geometric invariant theory offers the

tools to study the entire orbit structure of V, but a classification seems out of reach

for n>3 and d>4. We on the other hand pick specific forms P∈Vand study only

the orbit structure of ΩP, or even just the orbits that are of maximal dimension in a

component of ∂ΩP. In this chapter, we make some general observations and elaborate

on methods that we will apply in the chapters to come.

Unless explicitly stated otherwise, we only consider points P∈Vwhose stabilizer

GP={g∈GL(W)|P◦g=P}is reductive, mostly because it simplifies the theory

considerably and the condition is satisfied in our cases of primary interest, recall

Theorems 3.4.1 and 3.4.2.

We write ΩP:=P◦GL(W)⊆Vfor the orbit of Pand denote by ΩP⊆Vits closure

as in Definition 3.3.2. Recall also that by Theorem 3.3.3,ΩPis both the Euclidean and

the Zariski closure of ΩP. An element of ∂ΩPis called a degeneration of P.

5.1 Conciseness

Let us call a form P∈V=C[W]dconcise if it is not stabilized by any noninvertible

endomorphism, i.e.

∀a∈End(W):(P◦a=P)⇒(a∈GL(W)).

We will give other characterizations of this property here, for example it means that

there is no choice of coordinates C[W] = C[x1, . . . , xn]such that P∈C[x1, . . . , xn−1].

Informally, this may be expressed as “Puses all variables”. We will also see that being

concise is an open condition in Proposition 5.1.2.

5.1.1 Definition. Let x∈W∗=C[W]1and w∈W. We define the partial derivative of

xd∈C[W]din direction wto be the form ∂wxd:=d·x(w)·xd−1, c.f. [Lan12, eq. (2.6.6)].

By linear extension, this defines ∂wPfor any P∈C[W]dbecause C[W]d=SymdW∗

is spanned by powers of linear forms as a vector space [Mar73, Theorem 1.7]. If we

have chosen coordinates C[W] = C[x1, . . . , xn], we set ∂iP:=∂eiPfor 1 ≤i≤n, where

ei∈Wis the dual basis vector of xi∈W∗.

5.1.2 Proposition. Let Wbe a vector space of finite dimension. A form P∈C[W]dis

concise if and only if the linear map W→C[W],w↦→ ∂wPis injective.

Proof. The case d=0 is trivial, a constant form Pis stabilized by any endomorphism,

hence it is concise if and only if GL(W) = End(W), which is equivalent to W={0}.

The case d≥1 follows from Lemma 5.1.6 below for the identity map a=idW.

5.1.3 Remark. In particular, not being concise is a polynomial condition on the coeffi-

cients of P∈C[W]d: It is given by the vanishing of all maximal minors of the linear

map W→C[W]d−1,w↦→ ∂wP. Hence, the set of all concise forms is a Zariski open

subset of C[W]d.

5.1.4 Corollary. Let P∈C[x1, . . . , xn]d. Then, Pis concise if and only if the partial

derivatives ∂1P, . . . , ∂nPare linearly independent.

5.1.5 Remark. After choosing coordinates, a polynomial P∈C[x1, . . . , xn]satisfies

xi∈supp(P)if and only if ∂iP=0. Hence, Pis not concise if and only if there is

a choice of coordinates such that one of the variables does not appear in the support

of Pat all.

For a∈End(W), we denote by rk(a):=dim(a(W)) its rank.

5.1.6 Lemma. Let P∈C[W]dwith d≥1. For a∈End(W), we consider the linear

map δa:W→C[W],w↦→ ∂w(P◦a). The following conditions are equivalent:

(1) rk(a)≤rk(δa).

(2) For any b∈End(W)with P◦a=P◦b, we have rk(a)≤rk(b).

Proof. We choose coordinates C[W] = C[x1, . . . , xn]and set Q:=P◦a. Denote by

∇(Q):= (∂1Q, . . . , ∂nQ)∈C[x1, . . . , xn]1×n

d−1

the (row) vector of partial derivatives of Q. Let v1, . . . , vk∈Cnbe a basis of the vector

space {v∈Cn|∇(Q)·v=0}, so r:=n−kis the dimension of the C-vector space

spanned by the ∂iQ, i.e., r=rk(δa).

Assume that (2) holds. We have to show that rk(a)≤r. Let g∈GL(W)be an

invertible linear transformation that maps the i-th standard basis vector ei∈Cnto vi

for 1 ≤i≤k. Denoting by a dot the product of matrices, the chain rule yields

∂i(Q◦g) = ∇(Q◦g)·ei= (∇(Q)·vi)◦g=0 for 1 ≤i≤k.

This implies that Q◦g∈C[xk+1, . . . , xn]. We consider the linear map c∈End(W)

which maps xi◦c=0 for 1 ≤i≤kand xj◦c=xjfor j>k. Then, Q◦g◦c=Q◦g,

therefore P◦agcg−1=Q=P◦a. It follows that r=rk(c)≥rk(agcg−1)≥rk(a)by

our assumption.

Conversely, assume that (1) holds, so we have rk(a)≤r. Let b∈End(W)be such

that P◦a=P◦b. For any v∈ker(b), the chain rule yields

∇(P◦a)·v=∇(P◦b)·v= (∇(P)◦b)·b·v=0,

so any v∈ker(b)is a linear relation among the partial derivatives of P◦a. In other

words, k=dim(ker(δa)) ≥dim(ker(b)). Therefore,

rk(a)≤r=n−k≤n−dim(ker(b)) = rk(b).

5.2 Grading of Coordinate Rings and Projectivization

The next observation is that ΩPand ΩPare both affine cones in the following sense:

5.2.1 Definition. A subset C⊆Vis called an affine cone if ∀P∈C:C·P⊆C.

Note that GL(W)contains a central copy of Gm(C) = C×in the form of scalar

maps C×·idW, where idWdenotes the identity map. Given Q∈ΩP⊆C[W]dand

any t∈C, let ζ∈Cbe some d-th root of t. Then, tQ =ζdQ=Q◦ζidW∈ΩQ=ΩP.

Since scalar multiplication is continuous, we have the following general principle.

5.2.2 Lemma. If C⊆Vis an affine cone, then Cis also an affine cone.

Proof. The reductive group Gm= (C×,·)acts algebraically on Vby scalar multiplica-

tion. Denote by α:Gm×V→Vthe action morphism. Since Gm×C=Gm×C, we

have α(Gm×C)⊆α(Gm×C) = C. Hence, Cis Gm-invariant.

This means that ΩPis an affine cone. In particular, the coordinate ring C[ΩP]

inherits a N-grading from C[V]. On the other hand, C[GLn]has a natural Z-grading

that is the result of localizing at the SLn-invariant polynomial detn. The coordinate

ring C[ΩP] = C[GLn]GPinherits this grading. By Proposition A.2.9,

C[ΩP]d=⨁

λ∈Λ+

|λ|=d

V(λ)∗⊗V(λ)GP.

We will now see that under reasonable assumptions, C[ΩP]is also the localization

of C[ΩP]at some SL(W)-invariant and the grading of C[ΩP]is the same as the natural

one induced by this localization. We call a form P∈C[W]dpolystable if P◦SL(W)

is Zariski closed in C[W]d. Both the permanent and the determinant are polystable

[BI15, 2.9]. Furthermore, Bürgisser and Ikenmeyer proved:

5.2.3 Theorem ([BI15, 3.9]). If P∈V=C[W]dis polystable, then there exists a homo-

geneous invariant f∈C[ΩP]SL(W)such that C[ΩP] = C[ΩP][ f−1]. In particular,

ΩP={Q∈ΩP⏐⏐f(Q)=0}.

Furthermore, if fis homogeneous of minimal degree with this property, then fis

unique up to scalar.

Remark. Bürgisser and Ikenmeyer also study the minimal degree and other numerical

quantities related to these invariants, for the determinant and several other interesting

families of polynomials.

Studying the geometry of ΩPis equivalent to studying its projectivization P(ΩP),

which is a projective variety. Furthermore, we know that P(ΩP)is a smooth, open

subset of P(ΩP)which we understand well. While this is quite promising, it turns

out [BI15, 3.10,3.17,3.29] that ΩPis not a normal variety in most cases and in partic-

ular, when Pis detnor pernwith n≥3. It follows that in all these cases, ΩPhas

singularities. Since ΩPis smooth, these singularities will be part of the boundary

∂ΩP:=ΩP\ΩP, which is therefore a subvariety that needs to be analyzed further.

We note first that it is a hypersurface in our cases of interest:

5.2.4 Lemma. If GPis reductive, then ∂ΩPis pure of codimension one in ΩP.

Proof. The complement of an affine variety is always pure of codimension one [Gro67,

Corollaire 21.12.7]. The statement follows because for reductive GP, the orbit ΩPis

affine by Theorem A.1.9.

5.3 Rational Orbit Map

The orbit map ωP: End(W)→ΩPgiven by a↦→ P◦ais line-preserving, therefore it

can be viewed as a rational map

ϖP:PEnd(W)−→ PΩP(1)

[a]↦−→ [P◦a]

Since it can happen that P◦a=0 for a=0, the map ϖPis not defined everywhere.

If it was defined everywhere, it would be a dominant, projective morphism, therefore

surjective [Sha94, I.5.2]. We will transform ϖPinto a projective morphism by two

steps, the first of which is explained in this section. In Chapter 6we study the special

case where we obtain a morphism after the first step already. The second step is

postponed until Section 7.3 and we will see how to combine both steps in the final

two chapters.

5.3.1 Definition. For a form P∈C[W]d, we define

AP:={a∈End(W)|P◦a=0}(2)

and call it the annihilator of P. It is an affine cone in End(W)whose projectivization

is precisely the set of points where ϖP:PEnd(W)99K PΩPis not defined.

5.3.2 Remark. Another way to see the annihilator is as

AP={a∈End(W)|a(W)⊆Z(P)},

the set of all endomorphisms whose image is a linear subspace of the hypersurface

Z(P)⊆W. We will call a linear subspace L⊆Z(P)maximal if there is no other linear

subspace of Z(P)which properly contains L. With this notion, APis the set of all

a∈End(W)whose image is contained in some maximal linear subspace of Z(P).

The stabilizer GP={g∈GL(W)|P◦g=P}acts by left multiplication on End(W)

and ωPis GP-invariant with respect to this action, since ωP(g◦a) = P◦g◦a=P◦a

for all g∈GP. This action induces an action on PEnd(W)and consequently, the

rational map ϖPfrom (1) is invariant with respect to this induced action. We stress

again that GPis assumed reductive unless explicitly stated otherwise.

Consider now a map a∈End(W)with 0 ∈GP◦a, i.e., there is a sequence (gn)n∈N

with gn∈GPsuch that limn→∞gn◦a=0. Such points a∈End(W)are called

unstable with respect to the action of GPin the sense of geometric invariant theory

[MFK94], see also [New78;Kra85;Dol03].

5.3.3 Definition. Let Pbe a homogeneous form. The set of unstable endomorphisms

with respect to the action of GPis called the nullcone of Pand we denote it by

NP:={a∈End(W)⏐⏐0∈GP◦a}. (3)

See Subsection A.1.2 for other, equivalent definitions of the nullcone.

A central observation is the following:

5.3.4 Proposition. We have NP⊆ AP. In other words, if a∈End(W)is unstable with

respect to the left action of GP, then P◦a=0. This statement holds regardless of

whether or not GPis reductive.

Proof. Let a∈ NP, meaning 0 ∈GP◦a. Since ω−1

P(P◦a)is a closed subset of End(W)

containing GP◦a, it also contains its closure. Thus, 0 ∈GP◦a⊆ω−1

P(P◦a), which

means P◦a=P◦0=0.

We will now explain that the points in NPinduce “harmless” indeterminacies

of ϖP. We require the results from Subsection A.1.2, in particular the notion of

semistability. In our situation,

End(W)ss :=End(W)\NP={a∈End(W)⏐⏐0 /∈GP◦a}

is the set of semistable endomorphisms and PEnd(W)ss ⊆PEnd(W)is an open

subvariety of PEnd(W)which admits a categorical quotient, see Proposition A.1.11.

An important property of this quotient is the fact that

PEnd(W)ss//GP=Proj(C[End(W)]GP)

is a projective variety, even though PEnd(W)ss is only quasi-projective in general.

The first step for transforming ϖPinto a morphism is the following proposition,

which follows immediately from Propositions A.1.12 and 5.3.4.

5.3.5 Proposition. The domain of definition of ϖP:PEnd(W)99K PΩPis contained

in the open subset PEnd(W)ss. Furthermore, there is a commutative diagram

PEnd(W)ss

↓↓↓↓

ϖP→→PΩP

PEnd(W)ss//GP

φP

↗↗

and PEnd(W)ss//GPis a projective variety.

Remark. We can think of the rational map φPas having less indeterminacy than ϖP

because NPhas been removed before passing to the quotient.

For P∈C[W]d, the stabilizer GP⊆GL(W)acts on the vanishing set

Z(P):={w∈W|P(w) = 0}⊆W,

because P(w) = 0 implies P(g(w)) = (P◦g)(w) = P(z) = 0 for any g∈GP. In par-

ticular, GPacts on the set of linear subspaces of Z(P). We will often study this action,

in particular to determine the structure of GPin some cases. In light of Remark 5.3.2,

we therefore introduce the following terminology:

5.3.6 Definition. For a∈End(W), we denote by im(a):=a(W)its image. A linear

subspace L⊆Z(P)is called semistable if there is some a∈End(W)ss with L=im(a).

Otherwise, Lis called unstable.

5.3.7 Remark.

(1) For any a∈ NP, the space L:=im(a)is unstable: If some b∈End(W)satisfies

L=im(b), then there is a g∈GL(W)with b=a◦gand since gis an automor-

phism of End(W), we have 0 ∈GP◦a◦g=GP◦a◦g=GP◦b.

(2) Subspaces of unstable linear subspaces are unstable. Indeed, let a∈ NPand

L⊆im(a)a subspace. Then, L=im(a◦p), where p∈End(W)is some projection.

Since 0 ∈GP◦a◦p⊆GP◦a◦p, the space Lis also unstable.

Chapter 6

Closed Forms

If P∈C[W]dsatisfies ΩP=P◦End(W), we call it closed. This is equivalent to the

fact that ωPhas closed image.

6.0.1 Remark. If Pis closed, then ∂ΩPis always irreducible: It is the image under

ωP: End(W)→ΩPof the irreducible hypersurface S ⊆ End(W)of noninvertible

endomorphisms.

6.0.2 Example. Every quadratic form is closed. In fact, if there is some symmetric

matrix b∈Cn×nsuch that P(x) = xtbx, then for any endomorphism a, the form P◦a

is given by the matrix atba. We may assume that Pis concise, i.e. bis invertible.

It follows that{atba ⏐⏐a∈Cn×n}={ata⏐⏐a∈Cn×n}is the space of symmetric ma-

trices, hence P◦End(Cn) = C[x1, . . . , xn]2. In particular, this is equal to ΩPand Pis

closed.

6.1 A Sufficient Criterion

We know only very few examples of closed forms of higher degree. A classification

remains an interesting open problem, we can only give some partial results and pose

several questions. Recall the definitions of APand NPfrom Definitions 5.3.1 and 5.3.3.

A sufficient criterion for being closed is the following:

6.1.1 Proposition. Let P∈C[W]dbe a form with reductive stabilizer GP⊆GL(W). If

we have AP=NP, then Pis closed.

Proof. Recall the definition of ϖPfrom (1) and Proposition 5.3.5. The assumption

means that ϖPis defined everywhere on PEnd(W)ss and so the induced rational

map φP:PEnd(W)ss//GP→PΩPis in fact a morphism. Since it is dominant and

projective, its image is equal to PΩP. The statement follows because ϖPhas the same

image as φP.

6.1.2 Corollary. If the hypersurface Z(P)has no semistable linear subspaces, then P

is closed.

In Proposition 5.3.4, we saw NP⊆ AP. Proposition 6.1.1 states that Pis closed if

this inclusion is an equality. Having no counterexample but also no wide selection of

examples in general, we ask the following question:

6.1.3 Question. Does NP=APhold for every closed form P?

We know of no counterexample for Question 6.1.3, but for two examples of closed

forms that are already well-known, we illustrate that NP=AP.

6.1.4 Example. We know from Example 6.0.2 that concise quadratic forms are closed.

We claim that NP=APfor any such P∈V=C[x1, . . . , xn]2. We may assume

without loss of generality that P=x2

1+···+x2

nbecause any concise quadratic form

is an element of ΩP. On the one hand, we observe that

AP={a∈Cn×n⏐⏐P◦a=0}={a∈Cn×n|ata=0}.

The stabilizer of Pis the orthogonal group On={a∈GLn⏐⏐at=a−1}. As before,

we consider the operation of Onon Cn×nby left multiplication. By the First Fun-

damental Theorem for the Orthogonal Group [Pro06, 11.2, Theorem on p. 390], the

On-invariant functions on Cn×nare generated by the entries of the map a↦→ ata, and

by Lemma A.1.10 we therefore also have NP={a∈Cn×n⏐⏐ata=0}.

6.1.5 Example. Consider P:=z(y2+xz)∈C[x,y,z]3. The classification of ternary

cubic forms [KM02] implies that Pis closed. We will show that AP=NP. By

Example 2 in said reference, the stabilizer GP⊆GL3is generated by the matrices

uα:=⎛

⎜

⎝

1−2α−α2

0 1 α

0 0 1 ⎞

⎟

⎠and tε:=⎛

⎜

⎝

ε40 0

0ε0

0 0 ε−2⎞

⎟

⎠.

Note that U:={uα|α∈C}∼

=(C,+) is a normal unipotent subgroup of GPwhile

the group T:={tε|ε∈C×}∼

=(C×,·)is reductive. More precisely, tεuαtε−1=uε3α.

We have GP=TnU, the unipotent radical of GPis equal to Uand in particular GP

is not reductive and we cannot appeal to Proposition 6.1.1 to show that it is closed.

However, the definitions of NPand APdo not require Pto have reductive stabilizer.

Observe that Z(P) = Z(z)∪Z(y2+xz). The maximal linear subspaces of Z(P)are

therefore the plane Z(z)and any line in Z(y2+xz). We first note that Z(z)is unstable

because for any w∈Z(z), we have tε(w)→0 as ε→0. Therefore, we are left to show

that any w∈Z(y2+xz)\Z(z)spans an unstable line. After applying an appropriate

scaling matrix tε, we can assume that w= (−y2,y,1). Consider

gε:=tε−1◦uε3−y=1

ε4·⎛

⎜

⎝

1−2(ε3−y)−(ε3−y)2

0ε3ε3(ε3−y)

0 0 ε6⎞

⎟

⎠.

A straightforward computation shows that gε(w) = (−ε2,ε2,ε2)→0 as ε→0.

6.2 Normalizations of Orbit Closures

For almost every family of homogeneous forms studied in the context of GCT, the or-

bit closures are not normal varieties [Kum15;BI11]. In Section 4.3, we saw a prominent

example of an orbit closure whose normalization has a nice representation-theoretic

description: For this section, we choose coordinates C[W] = C[x1, . . . , xn]and con-

sider the n-form mnn:=x1···xn, the universal monomial. It is also the top symmet-

ric form. The variety Cn=Ω(mnn)is called the n-th Chow variety. In Chapter 4we

also used the notation Dn=Ω(detn), and we recall that Cn⊆Dnbecause mnnis the

restriction of detnto diagonal matrices.

By Lemma 4.3.1, the universal monomial is closed. Again, we observe that it

actually satisfies the condition we discussed in the previous section:

6.2.1 Proposition. We have Nmnn=Amnnfor all n≥2.

Proof. We fix n≥2 and let P:=mnn. We show that any a∈End(W)with P◦a=0

satisfies 0 ∈GP◦a. The group GPcontains all diagonal matrices t=diag(t1, . . . , tn)

with 1 =t1···tn. Assume that a∈Cn×n∼

=End(W)satisfies P◦a=0 and let

ai:=xi◦a=∑n

j=1aijxj, then we have 0 =P◦a=a1···an. Since C[W]is an integral

domain, it follows that there must be some iwith ai=0. Without loss of generality, we

may assume that a1=0, i.e., the first row of acontains only zeros. The diagonal matrix

tε:=diag(ε1−n,ε, . . . , ε)stabilizes Pand tε◦a=ε·a. We can view tεas a regular map

t:C→End(W)mapping ε↦→ tε. Since t(C×)⊆GP, we have t(C×)◦a⊆GP◦a.

Passing to the closure, we get 0 =t0◦a∈GP◦a.

The following theorem is well-known. It is implied by the proof of Lemma 4.3.3,

but we will also obtain it as a corollary of the upcoming Theorem 6.2.3 – we only

quote it here to provide context. Refer to Section A.3 for the notion of the polynomial

part V⊒0of a GL(W)-module V.

6.2.2 Theorem. Let ν: N(Cn)→Cnbe the normalization of the Chow variety. The

morphism νcorresponds to the inclusion C[N(Cn)] = C[Cn]⊒0⊆C[Cn].

Arguably, Theorem 6.2.2 and Remark 6.0.1 motivated Landsberg [Lan15] to ask

the following question: Is it true that whenever a GL(W)-orbit closure with reductive

stabilizer has an irreducible boundary, the coordinate ring of the normalization of the

orbit closure equals the polynomial part of the coordinate ring of the orbit?

The answer is negative, as we recently showed [Hü17]. We will instead prove that

C[N(ΩP)] = C[ΩP]⊒0is equivalent to AP=NP. Furthermore, we illustrate that

these equivalent conditions can be violated in cases where ∂ΩPis irreducible. The

main theorem of this chapter is the following, its proof is postponed to Section 6.3.

6.2.3 Theorem. Let Vbe a polynomial right-GL(W)-module where we denote the

action by V×GL(W)→V,(P,g)↦→ P◦g. Let P∈Vbe a point with reductive

stabilizer G⊆GL(W). Denote by Ω=P◦GL(W)its orbit and by ν: N(Ω)→Ωthe

normalization of its orbit closure.

There is a canonical, injective homomorphism ι:C[N(Ω)] ↪→C[Ω]⊒0of graded

C-algebras and GL(W)-modules, and the following statements are equivalent:

(1) The injection ιis an isomorphism.

(2) For all a∈End(W)with P◦a=0, we have 0 ∈G◦a.

(This means AP=NPin the case of homogeneous forms)

If either condition is satisfied, we have P◦End(W) = Ω.

Note that Theorem 6.2.3 implies Theorem 6.2.2 by Proposition 6.2.1. Furthermore,

we can use Theorem 6.2.3 to quickly deduce that the orbit of an elliptic curve is a

counterexample for the original question by Landsberg:

6.2.4 Proposition. Let W=C3and let P∈C[W]3be any form that defines a non-

singular curve in P2. Let ν: N(ΩP)→ΩPbe the normalization of ΩP. Then, the

stabilizer of Pis reductive, ∂ΩPis irreducible and C[N(ΩP)] is not isomorphic to

C[ΩP]⊒0as GL(W)-modules.

For the proof, we require some classical results about ternary cubics to deduce that

the stabilizer of Pis reductive and the boundary of its orbit irreducible:

6.2.5 Lemma. If P∈C[W]3defines a smooth curve, it has a finite (hence reductive)

stabilizer and ∂ΩP=ΩQis the orbit closure of a concise form Q. In particular, ∂ΩP

is irreducible.

Proof. By [KM02, Corollary 1], the stabilizer of Pis finite. Any finite group is re-

ductive by Maschke’s Theorem. A complete diagram of the degeneracy behaviour of

all ternary cubic forms can be found in Section 4 of [KM02]. Choosing coordinates

C[W] = C[x,y,z], the diagram implies that ∂ΩPis equal to the orbit closure of the

polynomial Q:=x3−y2z, regardless of the choice of P. One can compute that Qis

concise by means of Corollary 5.1.4. Hence, ∂ΩP=ΩQ. This variety is irreducible be-

cause it is the closure of the image of the irreducible variety GL(W)under the regular

map GL(W)→C[W]3,g↦→ Q◦g.

Proof of Proposition 6.2.4.By Lemma 6.2.5, the polynomial Phas a reductive stabilizer

and its orbit has an irreducible boundary. Let [w]∈P2be any point on the curve

defined by P, i.e., w∈Wis nonzero and P(w) = 0. Let a∈End(W)be of rank

one such that im(a)is spanned by w. Then, P◦a=0 and since GPis a finite

group, GP◦a=GP◦adoes not contain the zero map. Hence by Theorem 6.2.3,

the two GL(W)-modules C[N(ΩP)] and C[ΩP]⊒0are not isomorphic.

This counterexample also yields another important observation:

6.2.6 Corollary. Let S:=End(W)\GL(W)be the hypersurface of noninvertible endo-

morphisms. Then, ωP(S)⊆∂ΩPis not an irreducible component of ∂ΩPin general.

Proof. With Pas in Lemma 6.2.5, we know that Zariski almost every element of ∂ΩP

is concise, so the image of ωP(S)cannot be dense in ∂ΩP.

6.2.1 The Aronhold Hypersurface

As an example for Proposition 6.2.4, we consider the special case of the Fermat cu-

bic: Let W=C3,V=C[W]3=C[x,y,z]3and P:=x3+y3+z3∈Vfor the rest of

this subsection. We will write Ωand Ωinstead of ΩPand ΩPto simplify notation.

By Theorem 6.2.3 and Proposition 6.2.4, we know that the quotient C[Ω]⊒0/C[N(Ω)]

exists and that it is a nontrivial GL3-module, so it decomposes as a direct sum of

irreducible GL3-modules. We will explicitly compute some of the corresponding mul-

tiplicities.

Note that this is a special case as the orbit closure Ω⊆Vis a normal variety.

Indeed, this follows because Ωis the third secant variety of the 3-uple veronese em-

bedding P1→P3, which is projectively normal by [IK99, Thm. 1.56]. This simplifies

the calculation because we need not determine the normalization of Ω. Note that if P

is a generic regular cubic, Ωis not normal [BI15, Cor. 3.17 (1)].

Pdefines the elliptic curve with j-invariant equal to zero [Stu93, 4.4.7,4.5.8]. Its

orbit closure Ωis the hypersurface defined by the Aronhold invariant α∈C[V]4, see

[BI15, §3.2.1] for an explicit description. Thus, C[N(Ω)] = C[Ω] = C[V]/⟨α⟩. We

write in general

SymdSymmC3=⨁

λ∈Λ

V(λ)p(d,m;λ),

the coefficients p(d,m;λ)are also known as plethysm coefficients. In this case, we

have m=3.

The Aaronhold invariant αis a degree 4 polynomial and up to scaling, the unique

highest weight vector in C[V]of weight (4,4,4)with respect to the action of GL(W)∼

GL3(C). This means that the linear span of the GL3-orbit of αis isomorphic to the

irreducible GL3-module V((4,4,4)). We denote by Λ:=Λ+

3the set of dominant

weights of GL3(C), see Paragraph A.2.8. The degree 4 part of the homogeneous ideal

generated by αthen decomposes as

⟨α⟩d=C[V]d−4·α∼

=⨁

λ∈Λ

V(λ)p(d−4,3; λ−(4,4,4))

and we get

C[Ω]d∼

=⨁λ∈ΛV(λ)bλwhere bλ=p(d,3; λ)−p(d−4,3; λ−(4,4,4)).

aλbλmλλ

1 1 0 (4,2,0)

2 1 1 (6,0,0)

1 1 0 (4,4,1)

1 1 0 (5,2,2)

2 1 1 (6,3,0)

2 1 1 (7,2,0)

1 0 1 (8,1,0)

3 1 2 (9,0,0)

1 1 0 (6,4,2)

2 1 1 (6,6,0)

1 1 0 (7,3,2)

1 1 0 (7,4,1)

1 0 1 (7,5,0)

1 1 0 (8,2,2)

1 0 1 (8,3,1)

3 1 2 (8,4,0)

1 0 1 (9,2,1)

3 1 2 (9,3,0)

4 1 3 (10,2,0)

2 0 2 (11,1,0)

4 1 3 (12,0,0)

1 1 0 (6,6,3)

1 1 0 (7,6,2)

1 1 0 (8,4,3)

1 1 0 (8,5,2)

2 1 1 (8,6,1)

1 0 1 (8,7,0)

2 2 0 (9,4,2)

1 0 1 (9,5,1)

4 1 3 (9,6,0)

2 1 1 (10,3,2)

3 1 2 (10,4,1)

4 1 3 (10,5,0)

2 1 1 (11,2,2)

2 0 2 (11,3,1)

5 1 4 (11,4,0)

2 0 2 (12,2,1)

6 1 5 (12,3,0)

6 1 5 (13,2,0)

4 0 4 (14,1,0)

5 1 4 (15,0,0)

1 1 0 (6,6,6)

1 1 0 (8,6,4)

aλbλmλλ

1 1 0 (8,8,2)

2 2 0 (9,6,3)

1 1 0 (9,7,2)

1 0 1 (9,8,1)

1 0 1 (9,9,0)

1 1 0 (10,4,4)

1 1 0 (10,5,3)

3 2 1 (10,6,2)

3 1 2 (10,7,1)

4 1 3 (10,8,0)

1 1 0 (11,4,3)

3 2 1 (11,5,2)

4 1 3 (11,6,1)

3 0 3 (11,7,0)

1 0 1 (12,3,3)

4 2 2 (12,4,2)

3 0 3 (12,5,1)

9 2 7 (12,6,0)

3 1 2 (13,3,2)

5 1 4 (13,4,1)

7 1 6 (13,5,0)

3 1 2 (14,2,2)

4 0 4 (14,3,1)

9 1 8 (14,4,0)

3 0 3 (15,2,1)

9 1 8 (15,3,0)

1 0 1 (16,1,1)

9 1 8 (16,2,0)

5 0 5 (17,1,0)

7 1 6 (18,0,0)

1 1 0 (9,6,6)

1 1 0 (9,8,4)

1 1 0 (10,6,5)

1 1 0 (10,7,4)

2 2 0 (10,8,3)

2 1 1 (10,9,2)

2 1 1 (10,10,1)

2 2 0 (11,6,4)

1 1 0 (11,7,3)

3 2 1 (11,8,2)

2 0 2 (11,9,1)

2 0 2 (11,10,0)

1 1 0 (12,5,4)

aλbλmλλ

4 3 1 (12,6,3)

4 2 2 (12,7,2)

5 1 4 (12,8,1)

6 1 5 (12,9,0)

1 1 0 (13,4,4)

2 1 1 (13,5,3)

6 3 3 (13,6,2)

5 1 4 (13,7,1)

8 1 7 (13,8,0)

3 1 2 (14,4,3)

5 2 3 (14,5,2)

8 1 7 (14,6,1)

9 1 8 (14,7,0)

1 0 1 (15,3,3)

6 2 4 (15,4,2)

6 0 6 (15,5,1)

13 2 11 (15,6,0)

5 1 4 (16,3,2)

8 1 7 (16,4,1)

12 1 11 (16,5,0)

4 1 3 (17,2,2)

6 0 6 (17,3,1)

13 1 12 (17,4,0)

5 0 5 (18,2,1)

13 1 12 (18,3,0)

1 0 1 (19,1,1)

12 1 11 (19,2,0)

8 0 8 (20,1,0)

8 1 7 (21,0,0)

1 1 0 (10,8,6)

1 1 0 (10,9,5)

1 1 0 (10,10,4)

1 1 0 (11,8,5)

1 1 0 (11,9,4)

1 1 0 (11,10,3)

2 2 0 (12,6,6)

1 1 0 (12,7,5)

3 3 0 (12,8,4)

3 2 1 (12,9,3)

4 2 2 (12,10,2)

2 0 2 (12,11,1)

4 1 3 (12,12,0)

2 2 0 (13,6,5)

aλbλmλλ

2 2 0 (13,7,4)

4 3 1 (13,8,3)

5 2 3 (13,9,2)

5 1 4 (13,10,1)

4 0 4 (13,11,0)

4 3 1 (14,6,4)

4 2 2 (14,7,3)

7 3 4 (14,8,2)

7 1 6 (14,9,1)

9 1 8 (14,10,0)

2 1 1 (15,5,4)

6 3 3 (15,6,3)

7 3 4 (15,7,2)

9 1 8 (15,8,1)

11 1 10 (15,9,0)

2 1 1 (16,4,4)

4 1 3 (16,5,3)

10 3 7 (16,6,2)

10 1 9 (16,7,1)

15 2 13 (16,8,0)

4 1 3 (17,4,3)

8 2 6 (17,5,2)

12 1 11 (17,6,1)

14 1 13 (17,7,0)

2 0 2 (18,3,3)

9 2 7 (18,4,2)

10 0 10 (18,5,1)

20 2 18 (18,6,0)

7 1 6 (19,3,2)

11 1 10 (19,4,1)

17 1 16 (19,5,0)

5 1 4 (20,2,2)

9 0 9 (20,3,1)

19 1 18 (20,4,0)

7 0 7 (21,2,1)

17 1 16 (21,3,0)

2 0 2 (22,1,1)

16 1 15 (22,2,0)

10 0 10 (23,1,0)

10 1 9 (24,0,0)

. . . . . . . . .

Figure 6.2.1: Multiplicities in C[Ω]⊒0/C[N(Ω)] for the Fermat cubic, up to degree 8.

Note that bλcan be computed with the software package [BKT12].

Denoting by aλthe coefficients such that C[Ω]⊒0=⨁λ∈ΛV(λ)aλ, we are inter-

ested in the numbers mλ:=aλ−bλ, because:

C[Ω]⊒0/C[Ω] = ⨁λ∈ΛV(λ)aλ−bλ.

It follows from the Peter-Weyl Theorem (Theorem A.2.7) that aλ=dim(V(λ)G),

where G⊆GL3is the stabilizer of P. It is well-known [BI15, Prop. 2.4] that Gconsists

of permutation matrices and diagonal matrices whose diagonal entries are third roots

of unity. A matrix representation of the canonical projection V(λ)V(λ)Gwith

respect to the basis of semistandard Young tableaux (SSYT) is obtained by symmetriz-

ing each SSYT with respect to Gand straightening it [Ful97, § 7.4]. The quantity aλ

arises as the rank of this matrix. Using this method, we have computed the values of

the mλ:=aλ−bλup to degree 8, see Figure 6.2.1 on page 70.

6.2.7 Remark. A formula for aλis more involved than the one for bλ. Advancing

methods used in [BI11, Section 4.2] (see also [Ike12, Section 5.2]), Ikenmeyer [Ike16]

determined such a formula: For λ∈Λ, denote by |λ|:=λ1+λ2+λ3the sum of its

entries. We have aλ=0 unless|λ|=3dfor some d∈N. In this case,

aλ=∑

µ∈Λ

|µ|=d

∑

ν1,...,νd∈Λ,

|νk|=3·k·ˆ

µk

for all k

cλ

ν1,...,νd·

∏

k=1

p(ˆ

µk,3k;νk), (1)

where cλ

ν1,...,νddenotes the multi-Littlewood-Richardson coefficient and ˆ

µkdenotes the

number of times that kappears as an entry of µ.

6.3 Proof of Main Theorem

This section contains the proof of Theorem 6.2.3. We require results from Section A.3

and Subsection A.1.1.

Note that the stabilizer Gacts on the variety End(W)by multiplication from

the left. Since Vis a polynomial GL(W)-module, there is a well-defined morphism

ω: End(W)→Ω,a↦→ P◦a. For h∈Gand a∈End(W),

ω(ha) = P◦(ha) = P◦h◦a=P◦a=ω(a).

Therefore, ωis an G-invariant morphism. Since Gis a reductive algebraic group,

there is an affine quotient variety End(W)//Gtogether with a surjective morphism

π: End(W)→End(W)//Gand ωfactors as a morphism φ: End(W)//G→Ωsuch

that the following diagram commutes:

End(W)



↓↓↓↓

ω→→Ω

End(W)//G

↗↗(2)

Furthermore, the variety End(W)//Gis a normal variety because End(W)is normal

[TY05, 27.5.1].

The morphism φis a birational map. Indeed, End(W)//Gcontains GL(W)//G

as an open subset [TY05, 27.5.2] and the restriction of φto this open subset is the

isomorphism GL(W)//G∼

=Ω, [TY05, 25.4.6]. Since Ωis an open subset of its closure

[TY05, 21.4.3], this proves that φis generically one to one.

The normalization ν: N(Ω)→Ωis a surjective, finite morphism of affine alge-

braic varieties [GW10, Proposition 12.43 and Corollary 12.52]. By the universal prop-

erty of the normalization [GW10, Corollary 12.45] there exists a unique morphism

ψ: End(W)//G→N(Ω)which completes (2) to a commutative diagram:

End(W)



↓↓↓↓

ω→→Ω

End(W)//G

↗↗

ψ→→N(Ω)

↑↑↑↑



(3)

The morphism ψis dominant and therefore corresponds to an injective ring ho-

momorphism

C[N(Ω)] ⊆C[End(W)//G] = C[End(W)]G= (C[GL(W)]⊒0)G,

due to Lemma A.3.4. Taking G-invariants is with respect to the left action of Gon

C[GL(W)] and considering polynomial submodules is with respect to the right action

of GL(W)on C[GL(W)], so these two operations commute. Hence,

C[N(Ω)] ⊆(C[GL(W)]G)⊒0=C[Ω]⊒0.

As a consequence, the polynomial part of the coordinate ring of ΩPcan be identi-

fied with the ring of G-invariants in C[End(W)], where Gis the stabilizer of P.

6.3.1 Remark. There is a commutative diagram of GL(W)-equivariant inclusions of C-

algebras:

C[End(W)] C[Ω]

←←

↓↓

C[Ω]⊒0C[End(W)]G

↑↑

C[N(Ω)]

←←

(4)

Here, C[Ω]and C[End(W)]Ghave the same quotient field K. The inclusion

C[N(Ω)] ⊆C[End(W)]G=C[Ω]⊒0

is an inclusion of integrally closed subrings of K. (By definition, C[N(Ω)] is the

integral closure of C[Ω]in K.)

We now show that (1) of Theorem 6.2.3 implies P◦End(W) = Ω. Recall (3). The

condition C[N(Ω)] = C[Ω]⊒0holds if and only if the morphism ψis an isomorphism.

In this case it follows that φis the normalization of Ω. Thus, (1) implies in particular

that φis surjective and therefore ωis surjective, which means P◦End(W) = Ω.

We now ask when the inclusion C[N(Ω)] ⊆C[Ω]⊒0becomes an equality. We

will require an auxiliary lemma for the proof. The algebraic group C×=GL1acts

polynomially on a variety Xif the action morphism C××X→Xlifts to a morphism

C×X→X. We will denote this map by a dot, i.e., (t,x)↦→ t.x.

6.3.2 Lemma. Let Xand Ybe affine varieties, each of them equipped with polynomial

C×-actions admitting unique fixed points 0X∈Xand 0Y∈Y, respectively. Let

φ:X→Ybe a C×-equivariant morphism. Then, φ−1(0Y) = {0X}if and only if φis

finite.

Proof. The “only if” part is [Lan15, Lemma 7.6.3]. For the converse, assume that φis

finite. Let x∈Xbe such that φ(x) = 0Y. Then, φ(t.x) = t.φ(x) = t.0Y=0Yfor all

t∈C×and hence, C×.x⊆φ−1(0Y). But φ−1(0Y)is a finite set, therefore C×.xis finite

and irreducible, i.e., a point. This implies C×.x={x}, so xis a fixpoint for the action

of C×. It follows that x=0Xby uniqueness of the fixpoint.

Lemma 6.3.2 will be applied to the morphism φ: End(W)//G→Ω. We therefore

study the action of the scalar matrices C×⊆GL(W)on End(W)//Gand Ω. Observe

that the morphism φis equivariant with respect to this action. We need to make sure

that both varieties have a unique fixpoint in order to make use of Lemma 6.3.2.

We first reduce to the case where Vhas a unique fixpoint under the action of all

scalar matrices. For this purpose, fix some basis of W, so GL(W)∼

=GLnand let

V=⨁λ∈NnV(λ)be the decomposition of Vinto isotypical components, i.e., V(λ)is

a direct sum of irreducible modules of type λ. Note that the only weights λthat

appear are in Nnbecause Vis a polynomial GLn-module. Let P=∑λ∈NnPλbe the

corresponding decomposition of P, i.e., Pλ∈V(λ). Observe that the point ˜

P:=P−P0

has the same stabilizer as P, because any element of V0is GL(W)-invariant. Let

V:=⨁λ=0V(λ)be the complement of V0in V. Then, Ω∼

={P0}×Ω˜

P⊆V0×˜

V∼

and consequently, ΩP={P0}×Ω˜

P∼

=Ω˜

P. This shows that we may henceforth assume

V=˜

Vand P=˜

P. In this situation, the origin 0V∈Vis the only fixpoint under the

action of the scalar matrices. Consequently, it is also the only C×-fixpoint in Ω.

On the other hand, End(W)also has a unique fixpoint with respect to the left

action by scalar matrices, namely the zero map which we will denote by 0. At this

point, we require the following lemma to deduce that End(W)//Galso has a unique

fixpoint:

6.3.3 Lemma. Let Ebe an affine variety on which C×acts polynomially with a unique

fixpoint 0. Assume that a reductive group Gacts on Efrom the left such that the

actions of Gand C×commute. Then, the quotient E//Galso has a unique fixpoint

under the induced action of C×.

The proof of this lemma is slightly technical and will be given afterwards. Using

it, we conclude that π(0)is the unique fixpoint in X:=End(W)//Gand 0Vis the

unique fixpoint in Y:=Ω. The morphism φ:X→Ynow satisfies the conditions of

Lemma 6.3.2: We proceed to prove the equivalence of (1) and (2).

We first show (1)⇒(2). If (1) holds, ψis an isomorphism and φis a normalization

of Ω. Therefore, φis a finite morphism. By one direction of Lemma 6.3.2, this implies

that φ−1(0V) = {π(0)}. In other words, φ(π(a)) = 0Vimplies π(a) = π(0). We

have P◦a=ω(a) = φ(π(a)), so P◦a=0Vimplies π(a) = π(0), which is the same

as saying that the zero map 0 is contained in the closure of the G-orbit of a. This is

precisely (2).

For the converse implication, we assume (2). For any a∈End(W), the condition

0V=φ(π(a)) = ω(a)implies 0 ∈G◦aby (2). By construction of the GIT quo-

tient, this implies π(a) = π(0)and hence, φ−1(0V) = {π(0)}. The other direction

of Lemma 6.3.2 now states that φis a finite morphism. Any finite morphism is inte-

gral [GW10, Remark 12.10], so φis an integral birational map from a normal variety

End(W)//Gto Ω. By [GW10, Proposition 12.44], it follows that it is the normalization

of Ω, so ψis an isomorphism.

Proof of Lemma 6.3.3.We first note that 0 is a fixpoint for the action of Gas well. In-

deed, for any h∈Gand any t∈C×, we have t.h.0 =h.t.0 =h.0 because the actions

commute, so h.0 is a fixpoint for the action of C×. By uniqueness, this implies h.0 =0.

As h∈Gwas arbitrary, 0 is a fixpoint for the action of G.

We will denote by π:E→E//Gthe quotient morphism. Assume that x=π(a)is

any fixpoint of the action of C×. Observe that

{x}=C×.x=C×.π(a) = π(C×.a),

so C×.a⊆π−1(x). Since the action of C×is polynomial, the orbit map lifts to a

C×-equivariant morphism γ:C→Ewith γ(t) = t.afor t∈C×. Since

t.γ(0) = γ(t·0) = γ(0)

for all t∈C×, it follows that γ(0)is a C×-fixpoint in E, so γ(0) = 0. This implies that

0∈C×.a. Because 0 is a fixpoint for the action of G, the set{0}⊆Eis a closed G-orbit.

By the nature of the GIT-quotient, points of Ethat share a closed G-orbit are mapped

to the same point in the quotient. In this case, aand 0 share the closed orbit{0}and

it follows that x=π(a) = π(0). Thus, π(0)is the only fixpoint of the action of C×

on E//G.

Chapter 7

Techniques for Boundary Classification

We now turn to the more general case that P∈C[W]dis not closed, so there exist

forms in ∂ΩPthat are not of the form P◦afor any a∈End(W). Even so, any

degeneration of Pcan be obtained by approximation, as we will explain in Section 7.1.

In several examples, some components of ∂ΩPare orbit closures of degenerations

of P. In some cases, like the 3 ×3 determinant, every component of the boundary

contains a dense orbit – see Chapter 8. In fact, we have seen one such example

in Lemma 6.2.5. If a degeneration Qof Pis given, we can check whether ΩQis a

component of ∂ΩPby testing whether dim(GQ) = dim(GP) + 1. The latter implies

that ΩQis a proper, GP-invariant, codimension one subvariety of ΩP, hence disjoint

from ΩPand consequently a component of ∂ΩP. In Section 7.2, we describe a way to

compute the dimension of GQby means of linear algebra.

We can classify the components of ∂ΩPin some cases by constructing all of them

in this way. One way to ensure that all components have been found is to give a sharp

upper bound on the number of components of ∂ΩP. The upper bound we will use is

developped in Section 7.3.

7.1 Approximating Degenerations

Let C[t]be the polynomial ring over C. For an element q∈End(W)⊗C[t], we write

q=∑K

k=1qktkwith qk∈End(W), i.e., qkis the k-th coefficient of q. On the other hand,

one can also think of qas an endomorphism of the free C[t]-module W⊗C[t]. Any

form P∈C[W]can also be viewed as a polynomial with coefficients in C[t], giving

meaning to the notation P◦q.

7.1.1 Proposition. Let P∈C[W]d. There exists a natural number K∈Nsuch that for

every Q∈C[W]d, we have Q∈ΩPif and only if there exists a q∈End(W)⊗C[t]

such that P◦q≡tK·Q(mod tK+1).

Proof. We denote by CJtKthe ring of formal power series with complex coefficients.

Furthermore, we denote by C((t)) its fraction field, which consists of power series with

finitely many terms of negative degree. Both rings contain C[t]in a natural way. By

[LL89, Proposition 1 and the corollary on p. 11], there is a number M∈Nwith the

following property: For every Q∈ΩP, there exists q∈End(W)⊗C((t)) such that we

have P◦q≡Q(mod t)and tMq∈CJtK. We define K:=dM and claim that it has the

desired properties.

For the “⇐” direction, assume that ω(q) = P◦q=tKQ+∑k>KQktkwith cer-

tain polynomials Qk∈C[W]d. Let (ti)i∈Nbe a sequence of complex numbers that

converges to zero. For every i∈N, pick a d-th root ζiof tiand define the endo-

morphism ai:=ζ−K

iq(ti)∈End(W). Since Pis homogeneous of degree d, we have

P◦ζ−K

iidW=ζ−dK

iP=t−K

iPand we may conclude that

Pi:=P◦ai=t−K

i·(P◦q(ti)) = t−K

i(tK

iQ+∑

k>K

Qktk

i)=Q+∑k≥1Qk+Ktk

zero sequence

Hence, (Pi)i∈Nis a sequence of elements in ΩPwhich converges to Q. Theorem 3.3.3

implies that Q∈ΩP.

For the other direction, fix some Q∈C[W]dand let q=∑∞

k=−Mqktkbe such that

P◦q≡Q(mod t). We then define ¯

q:=∑K

k=0qk−Mtkand note that it is sufficient to

show

P◦¯

q≡tK·Q(mod tK+1). (1)

Recall that V=C[W]dand let E:=End(W). The orbit map ω:E→Vgiven by

ω(q) = P◦qis homogeneous of degree d, hence ω∈C[E]d⊗V. We can therefore

write ω=∑r

i=1ωi⊗Qiwith certain ωi∈C[E]dand Qi∈V.

Let p∈CJtKbe such that tM·q=¯

q+tK+1p. Then,

ω(q) = ω(t−M(¯

q+tK+1p))=t−K·ω(¯

q+tK+1p)

∑

i=1

t−K·ωi(¯

q+tK+1p)·Qi

≡

∑

i=1

t−K·ωi(¯

q)·Qi=t−K·ω(¯

q) (mod t).

Hence, Q≡P◦q=ω(q)≡t−K·ω(¯

q) = t−K·(P◦¯

q) (mod t). Multiplying this

equation by tKyields (1).

7.1.2 Remark. Note that for a particular Q, it can happen that the qfrom Proposi-

tion 7.1.1 is divisible by t, say q=trp. Then, we have

tKQ≡P◦q=P◦(trp) = tdr ·(P◦p) (mod tK+1)

so with k:=K−dr <K, we have tkQ≡P◦p(mod tk+1).

7.1.3 Definition. Let P,Q∈C[W]d. A power series q∈End(W)⊗C[t]which satisfies

P◦q≡tk·Q(mod tk+1)is called an approximation path from Pto Qand the num-

ber kis called its order. The minimum order of all approximation paths from Pto Q

is called the approximation order of Qwith respect to P.

The order of approximation of ΩPis defined to be smalles number Kthat can be

chosen in Proposition 7.1.1. It is also the maximum, taken over all Q∈ΩP, of the

approximation order of Qwith respect to P.

7.1.4 Example. We give an example from the well-known study of ternary cubic

forms, see [Kra85, I.7] for a complete classification. Compare also Lemma 6.2.5. We

consider the homogeneous form P=zx2−y3−z3∈C[x,y,z]3. It defines an irre-

ducible, nonsingular cubic curve and it is known that dim(GP) = 0. Let

q:=(1 0 0

0 0 0

0 0 0 )+(0 0 0

0 1 0

0 0 0 )·t+(0 0 0

0 0 0

0 0 1 )·t3=diag(1, t,t3)∈End(C3)⊗C[t].

Then, we have k=3 and

P◦q= (t3z)x2−(ty)3−(t3z)3=t3·(zx2−y3)−t9·z3

Note that Q:=zx2−y3is the equation of a cusp, which is singular. In particular we

have Q/∈ΩPand one can compute dim(GQ) = 1 with the methods of Section 7.2. It

follows that ΩQis an irreducible component of ∂ΩP.

Since the set End(W)⊗C[t]is entirely too large to choose from, we need more in-

formation on how to choose approximation paths. There are only two straightforward

observations we can make.

7.1.5 Remark. Let P∈C[W]d. An element Q∈ΩPhas approximation order zero

with respect to Pif and only if Q∈P◦End(W). In particular, Pis closed if and only

if ΩPhas order of approximation zero.

7.1.6 Remark. Let Q∈ΩPhave approximation order k≥1 and let q=∑K

i=0qitibe

an approximation path from Pto Qof order k. Then, P◦q0=0, because it is the

coefficient of tin P◦q. In other words, q0∈ AP.

We also ask the following question:

7.1.7 Question. Is the order of approximation of ΩPrelated to the degree of the

unique SL(W)-invariant from Theorem 5.2.3?

If P∈C[W]dand Q∈∂ΩP, we say that Qis a linear degeneration of Pif there is

alinear approximation path from Pto Q, i.e., one of the form b+at with b∈ AP. For

many cases we consider, only linear degenerations occur.

7.2 The Lie Algebra Action

The Lie algebra of an algebraic group Gis defined as the tangent space of Gat the

identity element 1 ∈G. We denote it by Lie(G):=T1(G). It carries the structure of a

Lie algebra, but this will not be of major importance for our considerations.

There is a strong relationship between the identity component G◦of an algebraic

group and its Lie algebra, c.f. [Hum98, 9] and [TY05, 23]. Moreover, there is the notion

of a Lie algebra action on a vector space, and the action of an algebraic group induces

an action of its Lie algebra as follows: The action of Gon a space Vis a morphism

ϱ:G→GL(V)of algebraic groups. Taking the derivative of this morphism at the

identity yields a linear map Dϱ:Lie(G)→Lie(GL(V)) ∼

=End(V). The action of

an element a∈Lie(G)on v∈Vreturns the vector Dϱ(a)(v)∈V. We also use the

common shorthands gl(V):=Lie(GL(V)) and gln:=Lie(GLn).

By Proposition A.1.3 Gis smooth, so we know that the dimension of Lie(G)as a

vector space is equal to the dimension of Gas a variety. We will describe the vector

space Lie(GP)explicitly in Corollary 7.2.3. Together with Theorem A.1.9, this also

provides a way to compute the dimension of ΩP.

Recall the partial derivative ∂wPof P∈C[W]din direction w∈Wfrom Defini-

tion 5.1.1.

7.2.1 Proposition. Let V=C[W]dand let GL(W)act on Vby precomposition. Then,

the induced action of gl(W) = Lie(GL(W)) = End(W) = W∗⊗Won Vis given by:

V×gl(W)−→ V

(P,a)↦−→ P∗a

where P∗(y⊗w) = y·∂wPfor rank one tensors.

Proof. Let ϱ: GL(W)→GL(V)be the morphism of algebraic groups corresponding

to the action of GL(W)on V. We have to compute the differential Dϱ:gl(W)→gl(V).

Define g(t):=idW+ (ty ⊗w), it satisfies g(0) = idWand g′(0) = y⊗w. Consider a

symmetric power xd∈SymdW∗=Vwith x∈W∗and observe

ϱ(g(t))(xd) = xd◦g(t) = xd◦(idW+ (ty ⊗w)) = (x+t·x(w)·y)d.

The coefficient of tin this expression is Dϱ(y⊗w)(xd). Expanding the right hand

side, we can see that the coefficient of tis equal to d·y·x(w)·xd−1=y·∂wxd. Since

symmetric powers span V, the result follows.

7.2.2 Corollary. If we choose coordinates C[W] = C[x1, . . . , xn], this gives an action of

GLnon V. The corresponding action of gln=Lie(GLn)in coordinates is given by

V×gln−→ V

(P,a)↦−→

∑

i=1

∑

j=1

aij ·xj·∂iP

where ∂iPdenotes the partial derivative of Pwith respect to xi.

Proof. After choosing coordinates, a matrix a= (aij)∈Cn×n∼

=glncorresponds to the

tensor a=∑n

i=1∑n

j=1aij ·(xj⊗ei), where ei∈Wis the dual basis vector of xi∈W∗.

The result follows from Proposition 7.2.1.

7.2.3 Corollary. Let V=C[x1, . . . , xn]dand let GLnact on Vby precomposition. For

P∈V, we have

Lie(GP)∼

={a∈Cn×n⏐⏐⏐⏐⏐

∑

j=1

∑

i=1(aij ·xj·∂iP)=0}

Proof. This follows from Corollary 7.2.2 and the fact that Lie(GP)is the set of all

a∈gl(W)that send Pto zero, see [TY05, 25.1.3].

7.2.4 Example. Let W:=Cd×d, so detd∈C[W]d. Let H:=G◦

detdbe the identity

component of the stabilizer of detd. We want to study Lie(H). For a square matrix

A∈W, we denote by tr(A):=∑d

r=1Arr its trace.

There is a bilinear map W×W→End(W)defined by (A,B)↦→ A⊗B, where

A⊗Bdenotes the map C↦→ ACBt. It is straightforward to check that this induces an

isomorphism W⊗W∼

=End(W). Let Eij ∈Wbe the matrix that has the entry 1 in

position (i,j)and zeros elsewhere. These matrices form a basis of Wand the matrix

representation of A⊗Bwith respect to this basis is the Kronecker product

(A⊗B)(ij),(rs)= (A⊗B)(Ers)ij = (A·Ers ·Bt)ij =AirBt

sj =AirBjs.

Let xij ∈W∗be the dual vector of Eij ∈W, so that C[W]is a polynomial ring

in the variables xij. Then detd=det(x)where x= (xij)is the matrix containing all

the variables. We denote by x♯the adjugate of the matrix x, i.e., the transpose of the

cofactor matrix of x. By definition, x♯

sr =∂rs detd. By Corollary 7.2.2, the action of

A⊗B∈End(W) = gl(W)on the form detdis given by

∑

i,j

∑

r,s

(A⊗B)(ij),(rs)·xij ·x♯

sr =∑

i,j,r,s

rixijBjsx♯

sr =tr(AtxBx♯).

In summary, we have A⊗B∈Lie(H)if and only if tr(AtxBx♯) = 0 for all x∈W.

Since GLd⊆Wis dense and S♯=det(S)·S−1for all S∈GLd, we conclude:

A⊗B∈Lie(H)⇐⇒ ∀S∈GLd: tr(AtSBS−1) = 0. (2)

Theorem 3.4.1 states that there is a surjective homomorphism of algebraic groups

SLd×SLdH, which induces sld×sldLie(H)

(A,B)↦→ A⊗B(A,B)↦→ (A⊗I) + (I⊗B).

Note that sld:=Lie(SLd)is equal to the vector space of d×dmatrices whose trace

vanishes [TY05, 19.1]. Clearly, (A⊗I) + (I⊗B)satisfies (2) when Aand Bare trace-

less, but it is not immediately obvious that any sum ∑r

i=1Ai⊗Bisatisfying (2) must

be of the form (A⊗I) + (I⊗B)with Aand Btraceless.

7.3 Resolving the Rational Orbit Map

Recall the rational map ϖPfrom (1) in Section 5.3. We denote by dom(ϖP)its domain,

i.e., the maximal open subset of PEnd(W)on which ϖis defined. There is a well-

known classical way to resolve the indeterminacies of a rational map: One considers

the graph

Γ:=Γ(ϖP) ={(a,ϖP(a))⏐⏐a∈dom(ϖ)}⊆PEnd(W)×PΩP

which has two natural morphisms β:Γ→PEnd(W)and γ:Γ→PΩPinduced by

the projections. By definition, Γis a projective variety. Since γis a dominant projective

morphism, it is surjective. With a good understanding of Γ, we could deduce a lot of

information about PΩP.

Unfortunately, while PEnd(W)is just a projective space, the variety Γand the

morphism βare not well-understood in general. A priori, we only know that βis a

blowup [Har06, II.7.17.3]. This is not very informative because any birational projec-

tive morphism is a blowup, see [Har06, II.7.17].

We want to give a minimal treatment here, even though one can define the blowup

of any scheme in a closed subscheme, see [GW10, pp. 406] and [Har06, pp. 160]. For

a classical treatment in the language of varieties, see [Har95, pp. 80]. We do require a

hint of scheme language but want to avoid the technical overhead: Hence, we restrict

to the affine case and heavily rely on the fact that blowups are local: See [GW10,

Prop. 13.91(2)] and use that open immersions are flat [GW10, Prop. 14.3(4)].

7.3.1 Definition. Let Xbe an affine complex variety and I⊆C[X]an ideal. It cor-

responds uniquely to a subscheme ˆ

A:=Spec(C[X]/I)of X. Choose generators

I=⟨φ0, . . . , φr⟩and consider the rational map

φ:X−→ Pr

x↦−→ [φ0(x): . . . : φr(x)]

We denote by Γ:=Γ(φ)the closure of{(x,φ(x)) |x∈dom(φ)}⊆X×Pr, it is called

the graph of φ. The projection β:Γ→Xis called the blowup of Xwith center ˆ

A. Up

to an automorphism of Γ, this does not depend on the choice of the generators of I,

see [Hü12, Prop. 1.19].

Remark. We will usually denote by A ⊆ Xthe subvariety of Xcorresponding to the

radical ideal √I, i.e., Ais the support of ˆ

7.3.2 Definition. In line with the above definition, we denote by ˆ

AP⊆End(W)the

closed subscheme of End(W)which corresponds to the ideal generated by the coef-

ficients of P◦a. Furthermore, we denote by ˆ

Ass

Pthe closed subscheme of the variety

End(W)ss which is defined locally by the same equations as ˆ

AP.

Note that ˆ

APis supported on AP={a∈End(W)|P◦a=0}. The graph of ϖPis

the projectivization of the blowup of End(W)with center ˆ

AP.

7.3.3 Example. In general, we do not have AP=ˆ

AP, i.e., the scheme ˆ

APis not

reduced. Consider W=C3and let P∈C[W]3be any regular cubic. The affine

cone Z(P)⊆Wis an irreducible surface of degree 3, therefore it cannot contain any

linear space of dimension two. It follows that the maximal linear subspaces of Z(P)

are the lines contained in this cone and any nonzero a∈ APis a rank one linear map

whose image is spanned by a point w∈Z(P). Hence, under the canonical isomor-

phism E:=End(W) = W⊗W∗, we have a=w⊗xfor some x∈W∗. The projective

class of ais a point in the segre embedding

P2×P2∼

=PW×PW∗⊆P(W⊗W∗) = PE

whose first coordinate is a point on the projective smooth curve defined by P, i.e., we

have PAP=PZ(P)×P2. In particular, PAPis a smooth variety.

However, ˆ

APis not even a variety: On the one hand, the coefficients of P◦aare

homogeneous polynomials of degree 3, so the ideal I⊆C[End(W)] generated by

them cannot contain any degree two polynomial. On the other hand, let µ∈C[E]2

be any 2 ×2 minor, then µvanishes on AP=Z(I)and therefore µ∈√I. Since I

contains no element of degree 2, we also have µ/∈I. Hence, I=√I, so ˆ

AP=AP.

Note also that passing to semistable points does not change this fact: In this par-

ticular example, the stabilizer of Pis a finite group by Lemma 6.2.5 and so every point

is semistable. Thus, Ass

P=ˆ

Ass

7.3.4 Remark. Example 7.3.3 generalizes as follows: Let n:=dim(W)and P∈C[W]d.

The hypersurface Z(P)cannot contain a linear subspace of dimension n−1, so any

endomorphism a∈ APsatisfies rk(a)<n−1. Therefore, the ideal of ˆ

APcontains

some power of the (n−1)×(n−1)minors, which are polynomials of degree n−1.

The coefficients of P◦aare homogeneous polynomials of degree din the entries of a,

so the ideal generated by them is not radical when d≥n.

If Pis generic, then Z(P)is smooth. Assuming d>1, it can be shown [Sha94,

Exercise II.1.13] that every linear subspace of Z(P)has dimension at most n

2, so ˆ

APis

nonreduced already when d≥n

7.3.5. For the rest of this section, we set E:=PEnd(W)and G:=PGL(W)for brevity.

A modest invariant of the boundary ∂ΩPis its number of irreducible components.

We are specifically interested in giving an upper bound on this number. We have the

following commutative diagram:

EϖP→→PΩP

↖↖

↗↗↗↗

Define U:=β−1(G) = γ−1(PΩP)and let Z:=Γ\Ube its complement.

Claim. The number of irreducible components of Zis an upper bound on the number

of irreducible components of ∂ΩP.

Proof of Claim. By commutativity, U=β−1(G) = γ−1(PΩP)and the restriction of γ

to Uis an isomorphism onto PΩP. Since γis also surjective, it follows that

γ(Z) = γ(Γ\U) = γ(Γ\γ−1(PΩP)) = PΩP\PΩP=P∂ΩP. (3)

Since P∂ΩPhas the same number of irreducible components as ∂ΩP, the statement

follows.

Unfortunately, this number does not seem easier to estimate than the number of

components of ∂ΩPitself. We will see later that passing to semistable points helps

tremendously, which is what we will discuss next.

We give V:=End(W)⊗C[W]daGP-module structure by acting trivially on the

right tensor factor. The graph Γof ϖPadmits the canonical embedding

Γ:={([a],[P◦a]) |a/∈ AP}⊆PEnd(W)×PΩP⊆P(V)(4)

via the Segre map, see [Lan12, Def. 4.3.4.1] and the following discussion.

7.3.6 Lemma. The embedding (4) realizes Γas a GP-invariant subvariety of P(V).

With the language of Subsection A.1.2, it turns Γinto a linearized GP-variety.

Furthermore, Γss is the graph of the restriction of ϖPto PEnd(W)ss.

Proof. We define Γ′:={([a],[P◦a]) |a/∈ AP}. It is an open, dense, GP-invariant subset

of Γand therefore, Γ=Γ′is closed and GP-invariant.

For the second claim, first note that a tuple ([a],[Q]) ∈Γis semistable if and

only if a∈End(W)ss by definition of the group action. Hence, Γss is the intersection

of PEnd(W)ss ×PΩPwith Γ. Since Γss also contains Γ′by Proposition 5.3.4, an

elementary topological argument yields the statement.

7.3.7 Definition. A blowup of a smooth variety with a smooth irreducible center of

codimension at least two is called a smooth blowup.

Remark. The condition on the codimension is only to avoid pathologies. If the center

has codimension one, the blowup is an isomorphism [Har06, II.7.14]. Note also that

the center of a smooth blowup is always reduced, i.e., a variety [Eis94, Cor. 10.14].

The following is our main tool for bounding the number of components of ∂ΩP.

Restricting to semistable points will allow us to actually verify the assumption in

some applications.

7.3.8 Proposition. Let β:Γss →(PEnd(W))ss be the projection from the graph of the

restriction of ϖPto semistable points. If βfactors as a sequence

Γss =Yk

βk

−−−→ ··· β2

−−−→ Y1β1

−−−→ Ess

of ksmooth blowups, then ∂ΩPhas at most k+1 irreducible components.

The proof of Proposition 7.3.8 will finish this section. We will require the following

well-known result which can be found in [Har06, II.8.24]. Intuitively, it states that a

smooth blowup only spawns a single irreducible component.

7.3.9 Proposition. Let β:Γ→Xbe a smooth blowup with center A. Then, Γis a

smooth variety and β−1(A)is a smooth, irreducible, codimension one subvariety of Γ

outside of which βis an isomorphism.

Note also that Proposition 7.3.8 would be easier to verify if we replaced Ess by E,

because the latter is projective. Since Ess is usually not projective, neither is Γss and we

have to pass to a quotient to regain this property. We will discuss this process before

proceeding to prove Proposition 7.3.8.

7.3.10. Clearly, β:Γ→Eis GP-equivariant and γ:Γ→PΩPis GP-invariant, in

particular there is a unique morphism ˜

γ:Γss//GP→PΩPsuch that

Γss

↘↘

β→→

↓↓↓↓

Ess

ϖP

↓↓

Γss//GP˜

γ→→PΩP

(5)

commutes, by Proposition A.1.11.

7.3.11 Lemma. The morphism ˜

γin (5) is a birational, surjective morphism and its

restriction to ˜

U:=˜

γ−1(PΩP)is an isomorphism ˜

U∼

=PΩP.

Proof. Since ϖPand βare dominant, so is ˜

γ. The variety Γss//GPis projective, there-

fore ˜

γhas closed image, so it is surjective. The orbit map ϖPrestricts to a surjective

morphism PGL(W)→PΩP, and its graph is the open subset

U={([g],[P◦g]) |g∈GL(W)}⊆Γss.

Hence, βrestricts to am isomorphism U∼

=PGL(W)of GP-varieties. We note here

that ˜

U=π(U)and ˜

γ(˜

U) = γ(U) = PΩP, so ˜

γ:˜

U→PΩPis a well-defined surjecive

morphism. As PΩPis smooth, ˜

γis an isomorphism if and only if it is injective, by

Zariski’s Main Theorem [TY05, 17.4.6]. Let [Q]∈PΩP, then we are left to show that

the fiber ˜

γ−1([Q]) contains exactly one element. To see this, let g∈GL(W)be such

that Q=P◦g. Then,

γ−1([Q]) = π(β−1(ϖ−1([Q]))) = π(β−1(GP◦[g])).

Since βis an equivariant isomorphism, β−1(GP◦[g]) is an orbit. Since πis constant

on orbits, we are done.

Proof of Proposition 7.3.8.We define ˜

Γ:=Γss//GPand consider the diagram (5). We

recall that G=PGL(W)⊆PEnd(W) = Eand U=γ−1(PΩP) = β−1(G).

Set Z:=Γss \U. By Lemma 7.3.11, the morphism ˜

γ:˜

Γ→PΩPis birational,

surjective, and restricts to an isomorphism on ˜

U=˜

γ−1(PΩP) = π(U). We also

define the set ˜

Z:=π(Z).

Claim. We have ˜

γ(˜

Z) = P∂ΩP.

Proof of Claim. As every GP-orbit in U∼

=Gis closed, Property (G4) of the good quo-

tient πimplies that ˜

Uand ˜

Zare disjoint. Since πis surjective, ˜

Zis the complement

of ˜

Uin ˜

Γ. The claim follows as in (3) because ˜

γis surjective by Lemma 7.3.11.

As P∂ΩP=˜

γ(˜

Z) = ( ˜

γ◦π)(Z)isthe image of Zunder a morphism, we are left to

verify that Zhas at most k+1 irreducible components.

Recall that βis a composition

Γss =Yk

βk

−−−→ ··· β2

−−−→ Y1β1

−−−→ Y0=Ess

of ksmooth blowups. We set ˆ

βi:=β1◦. . . ◦βifor all 1 ≤i≤k. Furthermore, denote

by Z0:=Ess \Gthe set of projective classes of semistable endomorphisms that are

noninvertible. Since Ess is open in E, it follows that Z0is an irreducible hypersurface

in Ess. Let Zi:=ˆ

β−1

i(Z0)for 1 ≤i≤k. Since the complement of Zk⊆Yk=Γss is

equal to U=β−1(G), we have Z=Zk. We prove by induction on ithat Zihas at most

i+1 irreducible components, which is tautological for i=0 and proves our initial

claim for i=k. For the induction step, we assume 1 ≤i≤kand that Zi−1has at

most iirreducible components.

Claim. We have Ai⊆Zi−1⊆Yi−1, where Aiis the center of βi:Yi→Yi−1.

Proof of Claim. Assume for contradiction that Aiintersects ˆ

β−1

i−1(G). Since Aiis of

codimension two and β−1

i(Ai)is of codimension one by Proposition 7.3.9, the Fiber

Dimension Theorem [TY05, 15.5.4] implies that there is a point g∈Gsuch that ˆ

β−1

i(g)

has positive dimension. As all βjare surjective, this would mean that β−1(g)has

positive dimension, which is a contradiction.

Since Ai⊆Zi−1, we get that

Zi=β−1

i(Zi−1) = β−1

i(Zi−1\Ai)∪β−1

i(Ai).

By Proposition 7.3.9,β−1

i(Ai)is irreducible and β−1

i(Zi−1\Ai)∼

=Zi−1\Aihas at

most iirreducible components. Therefore, Zihas at most i+1 irreducible compo-

nents.

7.3.12 Remark. One can prove an even more refined statement than Proposition 7.3.8.

Under the assumptions of the proposition, Z:=Γss \β−1(G)is a GP-invariant closed

subset of Γss. The action of GPon Γss therefore permutes the finite set of irreducible

components of Z. Denoting by ℓthe number of orbits of this action, it follows from

the proof of Proposition 7.3.8 that the number of irreducible components of ∂ΩPis at

most ℓ+1.

Chapter 8

The 3by 3 Determinant Polynomial

As outlined in Chapter 3, the orbit closure and boundary of the determinant poly-

nomial is of particular interest for GCT. Yet, very few explicit results describing the

geometry are known in low dimension. In this chapter we give a description of the

boundary of the orbit of the 3 ×3 determinant. These results have been previously

published in [HL16].

We view det3as the polynomial

det3:=det (x1x2x3

x4x5x6

x7x8x9)∈C[x1, . . . , x9]3,

a homogeneous polynomial of degree 3 on the space W:=C3×3. As before, we denote

by V:=C[W]3the space of all homogeneous forms of degree 3 on W.

Our main result is a description of ∂Ω(det3)that answers a question of Landsberg

[Lan15, Problem 5.4]: The two known components are the only ones. In Section 8.1 we

explain the construction of the two components. Our contribution lies in Section 8.2

where we show that there is no other component.

8.0.1 Theorem. The boundary ∂Ω(det3)has exactly two irreducible components:

• The orbit closure of the determinant of the generic traceless matrix, namely

Q1:=det ⎛

⎜

⎝

x1x2x3

x4x5x6

x7x8−x1−x5

⎞

⎟

⎠;

• The orbit closure of the universal homogeneous polynomial of degree two in three

variables, namely

Q2:=x4·x2

1+x5·x2

2+x6·x2

3+x7·x1x2+x8·x2x3+x9·x1x3.

Remark. The two components are different in nature: Ω(Q1)⊆det3◦End(W)is the

orbit closure of a polynomial in only eight variables; the second component is more

subtle and is not contained in det3◦End(W). This component has analogues in higher

dimension and some results are known about them [LMR13].

8.1 Construction of Two Components of the Boundary

We denote by ω: End(W)→Ω(det3)the orbit map ω(a) = det3◦a. Recall also the

description of Gdet3from Theorem 3.4.1.

8.1.1 Lemma. We have dim(Ω(det3)) = 65 and dim(Ω(Q1)) = dim(Ω(Q2)) = 64.

Proof. The stabilizer Gdet3has dimension 16 by Theorem 3.4.1, thereby it follows

that dim(Ωdet3) = 81 −16 =65 by Theorem A.1.9.

The dimension of Ω(Qi)for i∈{1, 2}can be deduced from its Lie algebra. By

Corollary 7.2.3, this amounts to computing the rank of a 165 ×81 matrix, which is

easy using a computer, see Program 8.1.

1from sympy import *

3def OrbitDimension(P,X):

4P = Poly(P,X)

5Df = [ Poly(x)*diff(P,y) for xin Xfor yin X ]

6Mn = list(set(sum((Poly(Q).monoms() for Qin Df),[])))

7return Matrix([[ Q.coeff_monomial(m) for Qin Df ]

8for min Mn ]).rank()

10 x,y,z,a,b,c,d,e,f = X = symbols(’xyzabcdef’)

12 Q1 = Matrix([[x,y,z],[a,b,c],[d,e,-x-b]]).det()

13 Q2 = a*x**2+b*y**2+c*z**2+d*x*y+e*y*z+f*z*x

15 print(OrbitDimension(Q1,X))

16 print(OrbitDimension(Q2,X))

Program 8.1: Stabilizer computation with Python [Pyt;Sym].

8.1.2 Lemma. Ω(Q1)is an irreducible component of ∂Ω(det3).

Proof. Since det3is concise and Q1is not, we have Ω(Q1)⊆∂Ω(det3). Lemma 8.1.1

implies that Ω(Q1)has codimension one in Ω(det3)and must therefore be an irre-

ducible component of ∂Ω(det3).

8.1.3 Lemma. Ω(Q2)is an irreducible component of ∂Ω(det3), distinct from Ω(Q1).

Proof. It is easy to see that Q2is concise whereas Q1is not, so Ω(Q1)contains no

concise polynomial, but Ω(Q2)does. It follows that the two orbit closures are distinct.

Let

b:=(0x1−x2

−x10x3

x2−x30)and a:=(2x6x8x9

x82x5x7

x9x72x4).

Th entries of these matrices are linear forms on W, so a,b∈End(W)with bpro-

jecting onto the space of antisymmetric matrices and aprojecting onto its orthogonal

complement of symmetric matrices.

Recall Section 7.1. To show that Q2∈∂Ω(det3), we will use the approximation

path b+at as outlined. It is clear that b∈ Adet3. The coefficient of tin det(b+ta)

is equal to tr(b♯a)by Jacobi’s formula, where b♯is the adjugate matrix of b. Fur-

thermore, we have b♯=utuwith u= (x3,x2,x1). Since tr(b♯a) = uaut=2Q2, we

have [Q2]∈PΩ(det3)by Proposition 7.1.1. Lemma 8.1.1 implies that Ω(Q2)has the

same dimension as ∂Ω(det3), so it is one of its irreducible components.

Note that Lemma 8.1.3 generalizes to higher dimensions: the limit of the deter-

minant on the space of skew-symmetric matrices always leads to a component of the

boundary of the orbit of detd, when d≥3 is odd, as shown by Landsberg, Manivel,

and Ressayre [LMR13, Prop. 3.5.1].

8.2 There Are Only Two Components

Throughout this section, we will denote by G:=Gdet3the stabilizer group of det3.

Recall the contents of Section 5.3. The annihilator A:=Adet3is precisely known,

thanks to the classification of the maximal linear subspaces of Wcontaining only

singular matrices [Atk83;FLR85;EH88]. These spaces are precisely the maximal

linear subspaces of Z(det3), see also Remark 5.3.2.

For every a∈ A, there is a h∈G◦such that im(h◦a)is a subset of one of the

following spaces of singular matrices:

(∗ ∗ ∗

∗ ∗ ∗

000),(∗ ∗ 0

∗ ∗ 0

∗ ∗ 0),(0 0 ∗

0 0 ∗

∗ ∗ ∗)and (0α−β

−α0γ

β−γ0)for α,β,γ∈C.

The first three are called compression spaces, we denote them by L1,L2and L3. The

fourth is the space of 3 ×3 skew-symmetric matrices, which we will denote by L0. Let

us write

End(W,L):={a∈End(W)|im(a)⊆L}(1)

for a linear subspace L⊆W. The sets Ei:=G◦◦End(W,Li)constitute the four

irreducible components of A. For example,

E0={[a]∈P(E)|∃S,T∈SL3: im(a)⊆S·L0·T}.

Indeed, they are irreducible because they are the image of the irreducible variety

G◦×End(W,Li)under the action morphism. Also, the Eiare not contained in one

another and Ais their union.

We denote by Ess

ithe set of semistable endomorphisms in Ei.

8.2.1 Lemma. We have Ass =Ess

0=∅.

Proof. Let gt:=diag(t,t,t−2), then gt∈SL3for all t∈Cand

gt·(∗ ∗ ∗

∗ ∗ ∗

000),(∗ ∗ 0

∗ ∗ 0

∗ ∗ 0)·gtand g−1

t·(0 0 ∗

0 0 ∗

∗ ∗ ∗)·g−1

all tend to 0 when t→0, for any constants ∗. This proves that the first three compo-

nents do not meet End(W)ss. To show that Ass is not empty, we exhibit an invariant

function f∈C[End(W)]G, which does not vanish at a point b∈End(W,L0). Pick any

three matrices A,B,C∈Wand let ˜

f: End(W)→Cbe the regular function

f(a):=tr(a(A)·a(B)♯·a(C)·a(A+B+C)♯).

The dot here is multiplication of 3 ×3 matrices. This is a G◦-invariant polynomial of

degree 6. Indeed, if h∈G◦is the map A↦→ SAT for some S,T∈SL3, then

f(h◦a) = tr(S·a(A)·T·T♯·a(B)♯·S♯·S·a(C)·T·T♯·a(A+B+C)♯·S♯)=˜

f(a)

because S♯=S−1and T♯=T−1. It follows that f:=˜

f+ ( ˜

f◦τ)is G-invariant,

where τ∈End(W)is the transposition map τ(A):=At. We now provide a point

b∈End(W)with f(b)=0. Let bbe the projection b:WL0which has the following

description in coordinates:

b=(0x1−x2

−x10x3

x2−x30). (2)

For generic choices A,B,C∈W, a simple computation shows that f(b)=0.

1from sympy import Matrix

3A = Matrix([[1,1,1],[1,1,1],[1,1,1]])

4B = Matrix([[1,0,1],[0,1,0],[1,0,1]])

5C = Matrix([[0,0,1],[0,1,0],[1,0,0]])

7b = lambda A: (lambda x: Matrix(

8[[ 0 , x[0], -x[1] ],

9[ -x[0], 0 , x[2] ],

10 [ x[1], -x[2], 0 ]]) )(list(A))

12 t = lambda a: ( a(A) *(a(B).adjugate()) *a(C) *

13 (a(A+B+C).adjugate()) ).trace()

14 f = lambda a: t(a) + t(lambda A: a(A).transpose())

15 print(f(b))

The above Python program [Pyt;Sym] evaluates fon bfor a particular choice of

matrices A,Band Cgiving f(b) = 2.

8.2.2 Lemma. An endomorphism a∈ Ass satisfies rk(a)≥3.

Proof. In [BD06, Thm. 2 and discussion above it], Bürgin and Draisma show that

any 2-dimensional subspace of Econtaining only singular matrices is contained in a

compression space. Therefore, if the image of ahad dimension 2 or less, then awould

lie in the nullcone, which is contrary to the choice of a.

Recall Section 7.3, in particular Definition 7.3.1. We set ˆ

Ass :=ˆ

Ass

det3, so Pˆ

Ass

are the indeterminacies of ϖ:PEnd(W)→PΩ(det3). The following proposition in

conjuction with Proposition 7.3.8 implies that ∂Ω(det3)has at most two irreducible

components, both of which we have described in Section 8.1. This finishes the proof

of Theorem 8.0.1.

8.2.3 Proposition. The subscheme Pˆ

Ass ⊆PEnd(W)is a smooth subvariety. In par-

ticular, the projection β:Γss →PEnd(W)ss is a smooth blowup, where Γss denotes

the graph of ϖrestricted to PEnd(W)ss.

Proof. We set E:=End(W)to shorten notation. By Definition 7.3.1 and Lemma 7.3.6,

Γss is the blowup of PEss along Pˆ

Ass. Therefore, we have to check that Pˆ

Ass is

irreducible and smooth, the latter also implies that it is reduced, i.e., ˆ

Ass =Ass.

Since scalar matrices stabilize any point in P(V), the stabilizer of [det3]∈P(V)is

the group H:=G◦CidW⊆GL(W).

Claim. Let b∈ Ass be the point defined in (2). We claim Ass =H◦◦b◦GL(W), the

orbit of bunder the action of H◦×GL(W)by multiplication from left and right.

Proof of Claim. For “⊇”, note that the left hand side is invariant under both actions

and contains b. Conversely, let a∈ Ass. By Lemma 8.2.1 and because L0is invariant

under transposition, we may assume that the image of ais included in L0, up to

replacing aby another point in its orbit H◦◦a. By Lemma 8.2.2, we also know that

rk(a)≥3, so asurjects onto L0. This implies that there is some g∈GL(W)such

that a=b◦g, and thus a∈H◦◦b◦GL(W).

In particular, Ass is irreducible, and smooth by Theorem A.1.9.(1). We will now

show that ˆ

Ass is also smooth. Since ˆ

Ass is invariant under the actions of both H

and GL(W)and its support Ass is an orbit under the same action, it suffices to verify

that ˆ

Ass is smooth at one single point, say b. This amounts to checking that the

dimension of the tangent space Tbˆ

Ass equals the dimension of Ass.

Recall that ˆ

Ais the scheme given by the coefficients of det3◦a=0, as polynomials

in the entries of a. By the Jacobian criterion [EH00, §V.3], we obtain the description

Tbˆ

Ass ∼

={c∈E⏐⏐⏐t2divides det3◦(b+tc)},

where tis a formal variable. The dimension of Tbˆ

Ass can be determined by means of

Program 8.2: It is equal to 35.

To calculate the dimension of Ass, we use the fact that it is an orbit under the

action of H◦×GL(W). More precisely, we consider the derivative of the orbit map

H◦×GL(W)−→ Ass

(a,c)↦−→ a◦b◦c

at the neutral element, yielding a surjective linear map

Lie(H◦)×gl(W)−→ TbAss

(a,c)↦−→ ab +bc

We have gl(W)∼

=E. Recall the Lie algebra of Gfrom Example 7.2.4. Similar to that

example, we can check that the Lie algebra of H◦is given by the maps (S⊗I) + (I⊗T)

for S,T∈W– i.e., there is no condition on Sand Tto be traceless because H◦contains

scalar matrices. We can therefore express the tangent space as

TbAss ={ab +bc |a∈Lie(H),c∈gl(W)}⊂TbE∼

={(S⊗I)◦b+ (I⊗T)◦b+b◦c|S,T∈W,c∈E}.

In other words, TbAss consists of all maps of the form

W−→ W

C↦−→ S·b(C) + b(C)·T+b(c(C))

for certain c∈Eand S,T∈W. Program 8.2 also verifies that this space has dimen-

sion 35, which terminates the proof.

1from sympy import *

2from sympy.abc import t

4b = lambda A: (lambda x: Matrix(

5[[ 0 , x[0], -x[1] ],

6[ -x[0], 0 , x[2] ],

7[ x[1], -x[2], 0 ]]) )(list(A))

9generic_matrix = lambda v,n: ( lambda s:

10 Matrix([ s[i::n] for iin range(n) ]) )(

11 symbols(’%s:%d’%(v,n*n)) )

13 A = generic_matrix(’a’,3)

14 c = generic_matrix(’c’,9)

16 cA = list(c*Matrix(list(A)))

17 cA = Matrix( [cA[i::3] for iin range(3)] )

18 cV = list(c)

20 D = (b(A) + t *cA).det()

21 L = Poly( diff(D,t).subs(t,0), list(A) ).coeffs()

22 L = [ collect(l,cV) for lin L ]

23 M = Matrix([[ term.coeff(i) for iin cV ] for term in L ])

25 # dim(End(W)) = 81

26 print("tangent space of indeterminacy:", 81-M.rank())

28 S = generic_matrix(’s’,3)

29 T = generic_matrix(’t’,3)

30 V = list(S) + list(T) + cV

31 B = S*b(A) + b(A)*T + b(cA)

32 L = [t for lin list(B) for tin Poly(l,list(A)).coeffs()]

33 N = Matrix([[ term.coeff(i) for iin V ] for term in L ])

35 print("tangent space of HxGL(W)-orbit:", N.rank())

Program 8.2: Tangent space computation with Python [Pyt;Sym].

8.3 The Traceless Determinant

A computation as in Lemma 8.1.2 shows that for d∈{3,4,5}, the orbit closure of the

traceless d×ddeterminant is a component of ∂Ω(detd). We will show in this section

that it is true for all d≥3. The main work in proving this claim is the following

description of the stabilizer of the traceless determinant. In fact, its dimension is

the only information required, but a complete description is certainly of independent

interest.

8.3.1 Theorem. Let W={A∈Cd×d⏐⏐tr(A) = 0}with d≥3 and let P∈C[W]dbe

the restriction of detdto W. The stabilizer group GPis reductive of dimension d2−1.

Moreover,

(1) The identity component of the stabilizer of Pis the group

G◦

P={g∈GL(W)⏐⏐⏐∃S∈SLd:∀A∈W:g(A) = SAS−1}.

(2) Let ∆⊆GL(W)be the group generated by

• the transposition map t:W→W,t(A) = At

• and all ζ·idWwhere ζ∈Cis a d-th root of unity.

Then, GP=∆·G◦

Pand GPhas|∆|=2dconnected components.

The proof is postponed briefly in order to first draw the announced conclusion.

8.3.2 Corollary. The orbit closure of the traceless determinant is an irreducible com-

ponent of ∂Ω(detd), for all d≥3.

For the proof of this corollary, we also require the following technical lemma,

which we will prove in direct succession.

8.3.3 Lemma. Let M∼

=Cn,D∈C[M]da concise form and W⊆Ma linear subspace

of codimension k. Let a∈End(M,W)be any projection onto W, let Q:=D◦aand

denote by P∈C[W]dthe restriction of Qto W. Then, dim(GQ) = dim(GP) + kn.

Proof of Corollary 8.3.2.We set M:=Cd×d,D:=detd∈C[M]dand W:=ker(tr). We

use the notation in Lemma 8.3.3, where k=1 and n=d2. By Theorem 8.3.1 we have

dim(GP) = n−1 and Lemma 8.3.3 states that dim(GQ) = n−1+n=2n−1. By

Theorems A.1.9 and 3.4.1 this implies dim(ΩQ) = dim(Ωdetd)−1.

Proof of Lemma 8.3.3.We can choose coordinates C[M] = C[x1, . . . , xn]such that the xi

with i≤n−kare coordinates on W. Then, Qis simply the polynomial that arises

from Dby setting xito zero for i>n−k, in the chosen coordinates. By Corollary 7.2.3,

Lie(GQ) ={a∈Cn×n⏐⏐⏐⏐⏐

∑

i=1

∑

j=1

aijxj∂iQ=0}={a∈Cn×n⏐⏐⏐⏐⏐

n−k

∑

i=1

∑

j=1

aijxj∂iQ=0}

because ∂iQ=0 for i>n−k. It suffices to show that dim(Lie(GQ)) = dim(GP) + kn.

We have

dim({a∈Cn×n⏐⏐⏐⏐⏐

n−k

∑

i=1

n−k

∑

j=1

aijxj∂iQ=0}) (3)

=dim({a∈C(n−k)×(n−k)⏐⏐⏐⏐⏐

n−1

∑

i=1

n−1

∑

j=1

aijxj∂iP=0})+ (n2−(n−k)2)

=dim(Lie(GP)) + 2kn −k2

We will show that⟨xj∂iQ⏐⏐1≤i≤n−k<j≤n,⟩Chas dimension k(n−k)and trivial

intersection with ⟨xj∂iQ|1≤i,j≤n−k⟩C. This implies that Lie(GQ)arises from (3)

by imposing k(n−k) = kn −k2additional linearly independent conditions, dropping

the dimension from dim(GP) + 2kn −k2to dim(GP) + kn as claimed.

The spaces intersect trivially because all monomials of polynomials in the second

space are divisibly by some xjwith j≤n−kand none of the monomials of polyno-

mials in the first space are.

As ais the identity on W, we have ∂iQ=∂iDfor i≤n−kand the ∂iDare linearly

independent by Proposition 5.1.2 because Dis concise. It follows that the xj∂iQwith

1≤i≤n−kand n−k<j≤nare all linearly independent.

8.3.1 Proof of Theorem 8.3.1

The thesis of Reichenbach [Rei16] contains a very detailed proof of Theorem 3.4.1, the

description of the stabilizer of detd. It can be adapted to verify Theorem 8.3.1, and

some parts of the proof require only slight modification or none at all. We therefore

refer to Reichenbach [Rei16] frequently and also make use of partial results therein.

We will denote by M:=Cd×dthe space of square matrices and recall that W⊆Mis

the subspace of traceless matrices. We will denote by Eij ∈Mthe matrix with entry 1

in position (i,j)and zeros elsewhere. We begin with the important observation that

we only have to show GP⊆Gdetd:

8.3.4 Lemma. Let g∈GP. If there are S,T∈GLdsuch that g(A) = S·A·Tfor all

A∈W, then there is a d-th root of unity ζ∈Csuch that T=ζ·S−1.

Proof. Set I:=TS. Since g∈GL(W), we have SAT ∈Wfor any A∈W. Hence,

∀A∈W: 0 =tr(SAT) = tr(ATS) = tr(AI).

Let Eij be the d×dmatrix with entry 1 in position (i,j)and zeros elsewhere. For

distinct indices i=j, we have 0 =tr(EijI) = Iij and therefore Iis a diagonal matrix.

Furthermore, from 0 =tr((Eii −Ejj)I) = Iii −Ijj it follows that all diagonal entries

of Iare equal to some ζ:=I11. Finally, pick some A∈Wwith det(A)=0. From

det(A) = det(g(A)) = det(SAT) = det(S)·det(A)·det(T) = det(TS)·det(A),

it follows that 1 =det(TS) = det(I) = ζd.

Recall that a singular subspace of Mis a linear space L⊆Msuch that det(A) = 0

for all A∈L. We denote by Lthe set of all maximal left ideals of Mas a ring. By

[Rei16, Satz 2.24], there is a bijection Pd−1∼

=Lgiven by v↦→ L(v), where

L(v):={A∈M|v⊆ker(A)}.

Here, vdenotes a line in Cd. In particular, every element of Lis a singular subspace

of M. For L∈ L, we set Lt:={At⏐⏐A∈L}, the image of Lunder transposition. The

set Lt={Lt⏐⏐L∈ L}is the set of all maximal right ideals of M, see [Rei16, §2.3] for

the complete classification. Every left ideal is the intersection of maximal left ideals,

and analogously for right ideals.

By [Rei16, Thm. 2.38], the set I:=L∪Ltis the set of all singular subspaces of M

of maximal dimension. We will write LW:={L∩W|L∈ L}and IW:=LW∪Lt

An important preliminary observation is the following proposition, which allows

us to argue essentially as in [Rei16]:

8.3.5 Proposition. The map I → IW,L↦→ L∩Wis a bijection. Furthermore, IWis

the set of all singular subspaces of Wof maximal dimension.

We require a sequence of lemmata which will be useful throughout the proof.

8.3.6 Lemma. Let L⊆Mbe a linear space and assume that S∈Msatisfies S·L⊆W.

If Eij ∈L, then Sji =0.

Proof. Let I:={(i,j)⏐⏐Eij ∈L}and C[M] = C[xij ⏐⏐1≤i,j≤d]. We consider the

matrix A:=∑(i,j)∈IxijEij ∈C[M]d×d. Then, 0 =tr(SA) = ∑d

i,j=1Sji Aij =∑(i,j)∈ISjixij

as a polynomial in C[M]. Hence, we have Sji =0 for all (i,j)∈Iby comparing

coefficients.

8.3.7 Lemma. Let L1,L2∈ L. Then, Wdoes not contain L1∩Lt

Proof. Assume for contradiction that I:=L1∩Lt

2⊆W. By [Rei16, Korollar 2.28], there

are S,T∈GLdsuch that L1·T=L2·Stare both equal to the space of all matrices

with vanishing first column. Since L1is a left ideal and Lt

2a right ideal,

L:=S·I·T=S·(L1∩Lt

2)·T= (S·L1·T)∩(S·Lt

2·T)

= (L1·T)∩(S·Lt

2) = (L1·T)∩(L2·St)t

is equal to the space of matrices where the first row and the first column vanish. In

particular, Eij ∈Lfor all i>1 and j>1. Since conjugation leaves the trace invariant,

(ST)−1·L=T−1IT ⊆W. By Lemma 8.3.6, this implies that (ST)−1has zero entries

everywhere except for the first row and column. Hence, rk((ST)−1)≤2<3≤d, a

contradiction because (ST)−1∈GLd.

8.3.8 Lemma. Wdoes not contain any nonzero left or right ideal of M.

Proof. Since Wis invariant under transposition, it suffices to show that Wdoes not

contain any left ideal of M. For contradiction, assume that L∈ L satisfies L⊆Wand

let rbe the maximum rank among elements of L, then r>0 because Lis nonzero. By

[Rei16, Satz 2.24], we have dim(L) = dr and by [FLR85, Theorem 2], a singular space

with this property admits matrices S,T∈GLdsuch that L′:=S·L·Tis the space

of all matrices whose first d−rcolumns vanish. Because r>0, we have Eid ∈Lfor

all i. Since (ST)−1·L′=T−1LT ⊆W, Lemma 8.3.6 implies that the last row of (ST)−1

vanishes, a contradiction.

Proof of Proposition 8.3.5.By definition, the map I → IW,L↦→ L∩Wis surjective. To

see that it is injective, assume for contradiction that there are I1,I2∈ I with I1=I2

and

J:=I1∩W=I2∩W.

By Lemma 8.3.8, we have dim(J) = d2−d−1. Furthermore, [Rei16, Lemma 2.29]

implies dim(I1∩I2)≤d2−2d+1. Lemmata 8.3.7 and 8.3.8 yield that I1∩I2is not

contained in W, providing the last inequality in

d2−d−1=dim(J) = dim(I1∩I2∩W)≤d2−2d.

This means d≤1, which is the contradiction we sought.

Let K⊆Wbe a singular subspace of maximal dimension. To finish the proof, we

have to show that K∈ IW.

Claim. There is some L∈ L with K⊆L.

Proof of Claim. Note that dim(K)≥d2−d−1 because Kis of maximal dimension and

for example, the singular spaces in IWhave dimension d2−d−1. There are two cases

to consider:

Case 1 (d>3). In this case, dim(K)>d2−2d+2. By [FLR85, Theorem 3], any

singular space K⊆Mwith this property is contained in a left or right ideal of M.

Case 2 (d=3). In this case, we have dim(K)≥9−3−1=5. Assume for contradic-

tion that Kis not contained in any maximal left or right ideal of M. As Kis singular,

it is contained in some maximal singular subspace of M, which we recall from Sec-

tion 8.2. Since dim(K)>3 and Kis not contained in any left or right ideal, Kis

contained in a space that is equivalent to

L:={(0 0 ∗

0 0 ∗

∗ ∗ ∗)}.

Since dim(L) = 5≤dim(K), in fact Kmust be equivalent to L. By definition, there

are matrices S,T∈GLdwith L=S·K·T. It follows that (ST)−1·L=T−1KT ⊆W

and Lemma 8.3.6 implies that the last row and column of the invertible matrix (ST)−1

vanish – a contradiction.

It follows from the claim that K⊆L∩W, the latter is a singular subspace of W.

Therefore, K=L∩Wbecause Kis maximal, hence K∈ IW.

With this at hand, we can mimic the six steps of the proof [Rei16, Thm. 4.2]. Recall

that we denote by t∈End(W)the transposition operator t(A):=At. The first step is

the following:

8.3.9 Proposition. Let g∈GP. After possibly replacing gby g◦t, the following holds:

For any K∈ LW, we have g(K)∈ LW.

Proof. We first note that for any K∈ IW, we have g(K)∈ IWbecause gmaps the

singular subspaces of Wto singular subspaces, and preserves the dimension.

Let K1,K2∈ LWand L1,L2∈ L with Li∩W=Ki. After possibly composing g

with t, we may assume that g(K1)∈ LW. Furthermore,

dim(g(K1)∩g(K2)) = dim(g(K1∩K2)) (gis injective)

=dim(K1∩K2) (gis injective)

=dim(L1∩L2∩W) (by definition)

=dim(L1∩L2)−1(Lemma 8.3.8)

=d2−2d−1. [Rei16, Lemma 2.29]

If we had g(K2)∈ Lt

W, then Lemma 8.3.7 and [Rei16, Lemma 2.29] would imply that

dim(g(K1)∩g(K2)) = d2−2d. Hence, we have g(K2)∈ LWas well. Since K2was

arbitrary, the statement follows.

For the rest of the proof, fix some g∈GP. We now construct two bijective maps

φ,ψ:Pd−1→Pd−1as follows: Note that gmaps any L(v)∩Wto some L(w)∩Wby

Proposition 8.3.9. Since LW∼

=L∼

=Pd−1, we can define φand ψvia

g(L(v)∩W) = L(φ(v)) ∩W,g(L(v)t∩W) = L(ψ(v))t∩W.

100

As gis a bijection, so are φand ψ. Moreover, φand ψare induced by semilinear

maps T:Cd→Cdand S:Cd→Cd, respectively. This follows from the upcoming

Lemma 8.3.10 and the fundamental theorem of projective geometry [Rei16, Theo-

rem 3.13].

8.3.10 Lemma. Both φand ψmap projective lines to projective lines.

Proof. Assume that u,v,w∈Pd−1are three pairwise different lines contained in

one plane, i.e., these projective points lie on the same projective line. Let L:=

L(u)∩L(v)∩L(w). Then, dim(L) = d2−2dby [Rei16, Satz 2.24, Korollar 2.27].

By Lemma 8.3.8, we have dim(L∩W) = d2−d−1. Thus,

d2−2d−1=dim(g(L∩W)) = dim(a(L(v)∩W)∩g(L(u)∩W)∩g(L(w)∩W))

=dim(L(φ(u)) ∩L(φ(v)) ∩L(φ(w)) ∩W).

Again by Lemma 8.3.8, we get that

d2−2d=dim(L(φ(u)) ∩L(φ(v)) ∩L(φ(w)))

and [Rei16, Satz 2.24] implies that the lines φ(u),φ(v)and φ(w)span a plane. The

argument for ψis completely analogous.

We have shown that g(L(v)∩W) = L(Tv)∩Wand g(L(v)t∩W) = L(Sv)t∩Wfor

all v. The third step is to show that Sand Tare linear maps, so we have S,T∈GLd.

The proof is word for word the same as in [Rei16, Thm 4.2,Proof Step 3], except that

the matrix E11 has to be replaced with E13, which is traceless. This is possible because

we assumed d≥3.

Let E:=E11 +···+Edd be the identity matrix, so M=W⊕(C·E). We extend g

to a map ¯

g∈GL(M)by declaring ¯

g(E):=Eand ¯

g|W:=g. We then define the map

u:M−→ M

A↦−→ St·¯

g(A)·T

Note that u∈GL(M)as well. For A∈L(v)∩W, we know g(A)·Tv =0, so in

particular we have u(A)v=St·g(A)·Tv =0, so u(L(v)∩W)⊆L(v). Equivalently,

one has u(L(v)t∩W)⊆L(v)t. We define

Dk:= (E11 +Ek1)−(E1k+Ekk)

for k>1, then{D2, . . . , Dd}∪{Eij |i=j}is a basis of W. Certainly, CEij for i=jcan

be expressed as an intersection of maximal left and right ideals of M. We observe that

for k>2, we can also write

CDk=⎛

⎝L(e1+ek)∩⋂

i/∈{1,k}

L(ei)⎞

⎠∩⎛

⎝L(e1−ek)t∩⋂

i/∈{1,k}

L(ei)t⎞

⎠.

101

It follows that u(CDk) = CDkand u(CEij) = CEij for i=jand k>2. Hence, there

are certain µk,µij ∈C×with u(Dk) = µkDkand u(Eij) = µijEij. We will show that all

these coefficients are identical:

8.3.11 Lemma. There is a µ∈C×such that µ=µkand µ=µij for all i=jand k>2.

Proof. For three distinct indices i,j,k(note that we require d≥3 here), the line

spanned by Eij +Ekj is contained in Wand can be expressed as the intersection of

maximal left and right ideals. Hence, there is some µijk ∈C×with

µijEij +µkjEkj =u(Eij +Ekj) = µijk ·(Eij +Ekj) = µijkEij +µijkEjk,

showing that µij =µijk =µkj. Similarly, one can show that µij =µik. It follows that

there is a µ∈C×with µ=µij for all i=j. We are left to show that µk=µholds

for all k>2. For this purpose, let j/∈{1, k}be a third index and observe that the line

spanned by Dk+Ej1−Ejk is contained in Wand can be expressed as the intersection

of maximal left and right ideals. Indeed, it is equal to

⎛

⎝L(e1+ek)∩⋂

i/∈{1,k}

L(ei)⎞

⎠∩⎛

⎝L(e1−ek)t∩L(e1−ej)∩⋂

i/∈{1,j,k}

L(ei)t⎞

⎠.

From this, we get certain νjk ∈C×with

µkDk+µEj1−µEjk =u(Dk+Ej1−Ejk) = νjk ·(Dk+Ej1−Ejk)

=νjkDk+νjkEj1−νjkEjk

and therefore, µk=νjk =µ.

We conclude that u=µ·idM. We can replace Sby µ−1Sand achieve that uis the

identity. Consequently, we are done by Lemma 8.3.4.

8.4 The Boundary of the 4×4Determinant

The next logical step would be to study det4and det5to accumulate a more solid

foundation of examples. The maximal linear subspaces of Z(det4)up to action of the

stabilizer are known [EH88, Cor. 1.3], [FLR85], they are as follows:

C0={(0∗ ∗ ∗

0∗ ∗ ∗

0∗ ∗ ∗)}and

C1={(∗ ∗ ∗ ∗

0 0 ∗ ∗

0 0 ∗ ∗)}

102

are compression spaces. The semistable spaces are the following:

L1={( 0α−β∗

−α0γ∗

β−γ0∗

0 0 0 ∗)⏐⏐⏐⏐⏐

α,β,γ∈C},

L2={( γ δ 0 0

0 0 γ δ

−α0−β0

0−α0−β)⏐⏐⏐⏐⏐

α,β,γ,δ∈C},

L3={(−β−δ0 0

α0−γ−δ

−δ0β0

γ α 0β)⏐⏐⏐⏐⏐

α,β,γ,δ∈C}.

By the computation given in Program 8.3, one can obtain the following result:

8.4.1 Proposition. For A,B∈Wwe let⟨A,B⟩:=tr(AtB). With respect to this bilinear

form, we choose orthogonal complements W=Li⊕Kifor i∈{1,2}.

There are surjective linear maps ai∈End(W,Li)and bi∈End(W,Ki)such that

ai+bi·tis an approximation path from det4to a polynomial Qi∈∂Ω(det4)and

Ω(Qi)is a component of ∂Ω(det4)for i∈{1, 2}.

Remark. The polynomial Q1is equal to

Q1=tr⎛

⎝(0x1−x2z3

−x10x3z2

x2−x30z1

0 0 0 t)♯

·(c e f 0

e b d 0

f d a 0

y3y2y10)⎞

⎠

=t·(a·x2

1+b·x2

2+c·x2

3+d·x1x2+e·x2x3+f·x1x3)

−(y1x1+y2x2+y3x3)(z1x1+z2x2+z3x3),

as a polynomial in C[t,a,b,c,d,e,f,x1,x2,x3,y1,y2,y3,z1,z2,z3]. Note that the first

polynomial from Theorem 8.0.1 appears with the factor tas the first summand here.

The polynomial Q2is equal to

Q2=tr⎛

⎝(x1x20 0

0 0 x1x2

x30x40

0x30x4)♯

·(y1y2b1b2

b3b4−y1−y2

y3a1y4a2

a3−y3a4−y4)⎞

⎠

=x2

1(x3a2−x4a1) + x2

2(x3a4−x4a3) + x2

3(b1x2−b2x1) + x2

4(b4x1−b3x2)

−(y1x2x3x4−x1y2x3x4+x1x2y3x4−x1x2x3y4),

as a polynomial in C[x1,x2,x3,x4,y1,y2,y3,y4,a1,a2,a3,a4,b1,b2,b3,b4].

Remark. As mentioned in [Lan15], these components were first observed by J. Brown,

N. Bushek, L. Oeding, D. Torrance and Y. Qi.

Remark. We used several approximation paths a+bt for det4with a∈End(W,L3).

Every such path gave a degeneration Qof det4whose orbit closure ΩQhad codimen-

sion one in ∂Ω(det4), i.e., ΩQwas not a component of the boundary.

103

2from sympy import *

3a,b,c,d,e,f,p,q,r,s,u,v,w,x,y,z = X = symbols(

4’ ’.join(c for cin ’abcdefpqrsuvwxyz’))

6def OrbitDimension(P,X):

7P = Poly(P,X)

8Df = [ Poly(x)*diff(P,y) for xin Xfor yin X ]

9Mn = list(set(sum((Poly(Q).monoms() for Qin Df),[])))

10 Mx = Matrix([[ Q.coeff_monomial(m) for Qin Df ]

11 for min Mn ])

12 return Mx.rank()

14 L1 = Matrix([

15 [ 0, x,-y, p ],

16 [-x, 0, z, q ],

17 [ y,-z, 0, r ],

18 [0,0,0,s]])

19 K1 = Matrix([

20 [ c, e, f, 0 ],

21 [ e, b, d, 0 ],

22 [ f, d, a, 0 ],

23 [u,v,w,0]])

24 Q1 = (L1.adjugate()*K1).trace()

26 L2 = Matrix([

27 [ p, q, 0, 0 ],

28 [ 0, 0, p, q ],

29 [ r, 0, s, 0 ],

30 [0,r,0,s]])

31 K2 = Matrix([

32 [ a, b, c, d ],

33 [ e, f,-a,-b ],

34 [ u, v, w, x ],

35 [ y,-u, z,-w ] ])

36 Q2 = (L2.adjugate()*K2).trace()

38 for Qin (Q1, Q2):

39 if OrbitDimension(Q,X)==(16*16)-(2*4*4-2)-1:

40 print("Found Component:\n%s" %str(Q.expand()))

Program 8.3: Computing degenerations of det4with Python [Pyt;Sym].

104

We note an interesting fact about the geometry of the spaces Ei:=G◦End(W,Li)

where G:=Gdet4, which might be related. From the proof of [EH88, Thm. 1.2], one

can deduce that (E2)ss is the disjoint union of Ess

2and Ess

3. In other words, Ess

3is the

complement of Ess

2in its closure, its “boundary” so to speak.

This leaves the following question:

8.4.2 Question. Does ∂Ω(det4)contain any irreducible component other than the or-

bit closures of Q1,Q2and the traceless determinant?

It is also noteworthy that L1has the space of skew-symmetric 3 ×3 matrices as its

“primitive part”, see [EH88] for a definition. We also ask if the construction of this

component generalizes:

8.4.3 Question. Let d=2k∈N. Let W:=Cd×dand consider the linear subspace

L:=

⎧

⎪

⎨

⎪

⎩

⎛

⎜

⎝

⎞

⎟

⎠

A∗

∗

0··· 0∗

⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐

A∈C(d−1)×(d−1),

A=−At

⎫

⎪

⎬

⎪

⎭

Clearly, Lis a singular subspace of detd.

We ask whether there exists a vector space complement W=L⊕Uand linear

maps a:WLand b:WUsuch that a+bt is an approximation path from detd

to a polynomial Qwhose orbit closure is an irreducible component of ∂Ω(detd).

We end with the sobering remark that the 5 ×5 case is already quite substantially

more difficult. The annihilator Adet5contains infinitely many orbits under the action

of Gdetd, as shown in [Bor+16, Theorem 5]. This means that for general d, the Gdetd-

orbit structure of Adetdis presumably quite involved. Understanding this structure is

more or less equivalent to the classification of all maximal linear subspaces of detd.

105

Chapter 9

The Binomial

It would be desirable to have a description of ∂Ω(Pd)for a complete family of poly-

nomials P= (Pd)d∈N. A trivial example is (mnd)d∈Nbecause it is closed, as noted in

Lemma 4.3.1. However, this is of little educational value. Our suggestion is to study

the polynomial family

bnd:=x1···xd+y1···yd, (1)

the generic binomial. We assume d≥3 throughout.

9.0.1 Remark. Note that bnd∈Ω(detd). For example,

bn5=det ⎛

⎜

⎝

x1y10 0 0

0x2y20 0

0 0 x3y30

0 0 0 x4y4

y50 0 0 x5

⎞

⎟

⎠

and this construction generalizes easily.

Furthermore, bndis also a special case of another important polynomial family,

namely the sums of products polynomial ∑k

i=1∏d

j=1xkj, see [CKW10, Chapter 11] and

[Kay12].

9.0.2. Here is a brief summary of our analysis of the binomial:

(1) Let S ⊆ End(W)be the set of noninvertible endomorphisms. The set B:=bnd◦S

is an irreducible component of ∂Ω(bnd). It is not an orbit closure.

(2) There is only one semistable linear subspace of Z(bnd). Using it, we construct

a linear degeneration Qof bndsuch that ΩQis an irreducible component of the

boundary.

(3) We prove that the scheme ˆ

Abndis generically smooth, in particular generically

reduced. If it is reduced and its components intersect transversally, then ∂Ω(bnd)

has exactly these two irreducible components.

We let W:=Cd×Cdwith coordinates xion the left factor and yjon the right one,

so we have bnd∈V:=C[W]d. It is easy to see that bndis concise.

107

9.1 Stabilizer and Maximal Linear Subspaces

We will denote by Hd:=Gbndthe stabilizer group of the binomial. The main goal of

this section is to prove the following result:

9.1.1 Theorem. Let W:=Cd×Cdwith d≥3. The stabilizer group Hd⊆GL(W)is

reductive of dimension 2d−2. Moreover,

(1) The identity component of the stabilizer of Pis the group

H◦

d={diag(s1, . . . , sd,t1, . . . , td)⏐⏐⏐∏d

i=1si=∏d

i=1ti=1}.

(2) Let K⊆GL(W)be the group generated by

• the map t:W→W,(v,w)↦→ (w,v), and

• the group Sd×Sd⊆GLd×GLd⊆GL(W)of permutation matrices that per-

mute the first and last dcoordinates among themselves, respectively.

Then, K∼

=Sd≀Z2and Hd=K·H◦

dhas|K|=2·d!·d! connected components.

9.1.2 Corollary. By Theorem A.1.9, dim(Ω(bnd)) = 4d2−2d+2.

One can prove Theorem 9.1.1 by means of the explicit description of Lie(Hd)from

Corollary 7.2.3, but another method is to study the action of Hdon linear subspaces

of the vanishing locus Z(bnd)⊆W=Cd×Cd. It is straightforward to provide linear

subspaces of Z(bnd): The most canonical choice are the spaces Lij :=Z(xi,yj)for

1≤i,j≤d. Furthermore, there is the space

L0:={(w1, . . . , wd,ζw1, . . . , ζwd)|w1, . . . , wd∈C}⊆W,

where ζ∈Cwill always denote a fixed d-th root of −1, i.e., we have ζd=−1. For the

proof of Theorem 9.1.1, we will prove the following proposition as an auxiliary result.

9.1.3 Proposition. The maximal linear subspaces of Z(bnd)consist of the Lij and the

spaces h(L0)for h∈Hd. Furthermore, the Lij are unstable.

Proof of Theorem 9.1.1.It is easy to see that the elements of H◦

dand Kstabilize bnd.

We are left to show that Hd⊆K·H◦

d. The elements of Hdact on the set of linear

subspaces of Z(bnd)of any fixed dimension, so by Proposition 9.1.3 they permute

the Lij. Dually, the action of Hdon W∗permutes the spaces L∗

ij :=⟨xi,yj⟩ ⊆ W∗for all

1≤i,j≤din the sense that for all h∈Hd, there are 1 ≤r,s≤dsuch that

L∗

ij ◦h:=⟨xi◦h,yj◦h⟩=⟨xr,ys⟩=L∗

rs.

108

Let h∈Hdand 1 ≤k≤d. We claim that there is an index 1 ≤i≤dsuch that the

linear form xk◦his either a scalar multiple of xior yi. To this end, let 1 ≤i,j,r,s≤d

be such that L∗

k1◦h=L∗

ij and L∗

k2◦h=L∗

rs, then

⟨xk◦h⟩=⟨xk⟩◦h= (L∗

k1∩L∗

k2)◦h=L∗

ij ∩L∗

rs.

Hence, the two planes L∗

ij and L∗

rs intersect in the line spanned by (xk◦h), in particular

this intersection is nontrivial. It follows that i=ror j=sbecause for i=rand j=s,

we have L∗

ij ∩L∗

rs ={0}. Therefore, either ⟨xk◦h⟩=⟨xi⟩or ⟨xk◦h⟩=⟨yj⟩as claimed.

In the same manner, we can show that yk◦his some scalar multiple of a coordinate

function. This means that his the product of a permutation and a diagonal matrix.

The result is a straightforward corollary.

We are left to show Proposition 9.1.3. The proof has been split up into the following

three lemmata:

9.1.4 Lemma. A maximal linear subspace L⊆Z(bnd)which is contained in a coordi-

nate hyperplane is equal to Lij for some choice of iand j.

Proof. Assume L⊆Z(xi). It follows that any (v1, . . . , vd,w1, . . . , wd)∈Lmust satisfy

w1···wd=0. Consequently, L⊆Li1∪···∪Lid. It follows that L=Lij for some j.

9.1.5 Lemma. The spaces Lij are unstable.

Proof. It is sufficient to show that L1dis unstable since Lij =h(L1d)for an appropriate

permutation h∈Hd. To see that L1dis unstable, let a:W→L1dbe any linear map

and consider

hε:=diag(ε1−d,ε, . . . , ε,ε1−d)∈Hd,

where ε1−dis in the first and last position. Note that the representing matrix of ahas

vanishing first and last row. Therefore, hε◦a→0 as ε→0.

9.1.6 Lemma. Let L⊆Z(bnd)be a maximal linear subspace which is not contained

in any coordinate hyperplane. Then, there is an h∈Hdsuch that h(L) = L0.

Proof. Since Lis not contained in any of the coordinate hyperplanes, there is a point

(u1, . . . , ud,v1, . . . , vd)∈Lsuch that ui=0 and vi=0 for all 1 ≤i≤d. Let u∈Cbe

such that ud=u1···ud. We apply the scaling matrix

diag(u

u1, . . . , u

ud,v2···vd

ζd−1ud−1,ζu

v2, . . . , ζu

vd)∈H◦

to the space Land thereby achieve that (u, . . . , u,ζu, . . . , ζu)∈L. By scaling, we

get that p:= (1, . . . , 1, ζ, . . . , ζ)∈L. We denote by Kthe discrete part of Hd, i.e.,

109

the finite group of permutations that stabilize bnd. We now claim that L⊆K.L0,

so Lis contained in a finite union of translates of L0. This will imply the state-

ment because Lhas to be equal to one of the translates of L0. For the proof, let

q= (u1, . . . , ud,v1, . . . , vd)∈Lbe any point different from p. Since the entire line

containing pand qmust be a subset of Land subsequently a subset of Z(bnd), this

means that bndvanishes at Tp −qfor any T∈C. In other words,

∏

i=1

(T−ui) +

∏

i=1

(ζT−vi) =

∏

i=1

(T−ui)−

∏

i=1

(T−ζ−1vi)

as an identity of polynomials in C[T]. Hence, ui=ζ−1viup to a permutation which

stabilizes bnd.

For future reference we consider the matrix

b:=(I0

ζI0), (2)

then End(W,L0) = b◦GL(W)is the set of all endomorphisms that map into L0. From

Proposition 9.1.3, we obtain:

Ass

bnd={h◦a|a∈End(W,L0)ss}=Hd◦End(W,L0)ss =Hd◦b◦GL(W)ss. (3)

9.2 The First Boundary Component

There are certain natural degenerations of bndwhich we will study here. We introduce

the notation P∼Q:⇔P∈ΩQand say that Pis linearly equivalent to Qin this case.

9.2.1 Remark. An important property of linear equivalence is the fact that P∼Q

implies GP∼

=GQ, indeed if Q=P◦gthen it is easy to see that GQ=g−1◦GP◦g.

For the rest of this section, we also set

S:={a∈End(W)|rk(a)<2d},

so Sis the hypersurface of noninvertible endomorphisms on W=C2d. Our main

result of this section is the following:

9.2.2 Proposition. Let B⊆Ω(bnd)be the closure of bnd◦S. Then, Bhas codimension

one in Ω(bnd)and it is an irreducible component of ∂Ω(bnd).

For the proof, we define the polynomials

Qr,s=x1···xd+y1···yd−1·(r

∑

k=1

xk+

∑

k=1

yk)

for r≤dand s<d. Note that Qr,s∈bnd◦S. We defer the proof of the following

auxiliary result:

110

9.2.3 Lemma. The polynomials Qd−1,d−1and Qd,d−1are elements of bnd◦S whose

orbits are disjoint. Furthermore, both orbits are of codimension 2 in Ω(bnd).

Proof of Proposition 9.2.2.By definition, Bis an irreducible closed subset of Ω(bnd),

because it is the closure of the image of the irreducible variety Sunder the orbit map.

Let us check that it is contained in ∂Ω(bnd): The elements of bnd◦S are not concise.

Since that is a closed condition by Remark 5.1.3, no element of Bis concise. As all

elements of Ω(bnd)are concise, we know that Bmust lie in the complement.

We are left to determine the dimension of B. By Lemma 9.2.3,Bhas codimension at

most 2. Assume for contradiction that the codimension of Bin ∂Ω(bnd)is equal to 2.

The orbit closure Ω(Qd,d−1)is an irreducible closed subset of Band it has the same

dimension as Bby Lemma 9.2.3 and our assumption. This implies that Ω(Qd,d−1) = B.

We have Qd−1,d−1∈B=Ω(Qd,d−1), but Qd−1,d−1/∈Ω(Qd,d−1)by Lemma 9.2.3. This

means Qd−1,d−1∈∂Ω(Qd,d−1). However, the orbit of Qd−1,d−1has the same dimension

as the orbit of Qd,d−1, which is a contradiction to Theorem A.1.9.(3).

It follows that the codimension of Bis at most 1, and since it is completely con-

tained in the boundary, this implies that it is a component.

Proof of Lemma 9.2.3.Let r∈{d−1, d}and Q:=Qr,d−1. We define the linear form

ℓ:=x1+···+xr+y1+···+yd−1, so Q=x1···xd+y1···yd−1·ℓ. Furthermore, we

denote by a∈ S the endomorphism that satisfies yd◦a=ℓand ais the identity on all

other coordinates, so that Q=bnd◦a. We will compute the dimension of GQand the

number of its connected components. We will see that the former does not depend

on r, whereas the latter does – this will imply the lemma by Remark 9.2.1. To obtain

a description of GQ, we study its action on the set of linear subspaces of Z(Q).

Claim. If L⊆Z(Q)is a linear subspace with dim(L) = 2d−2, then there are indices

1≤i,j≤dsuch that L=Z(xi,yj◦a).

For j<d, this means L=Lij =Z(xi,yj)and otherwise L=Z(xi,ℓ).

Proof of Claim. Since Q=bnd◦a, we know that a(L)is a linear subspace of Z(bnd), so

the space a(L)is contained in one of the spaces from Proposition 9.1.3. We first show

that there are 1 ≤i,j≤dwith a(L)⊆Lij.

The form yd−ℓvanishes on im(a), but it is straightforward to verify that it does

not vanish on h(L0)for any h∈Hd. Therefore, a(L)=L0, but it might be properly

contained in it. We show that this is not possible: Since d≥3, we have

dim(a(L)) ≥dim(L)−1=2d−3≥d=dim(L0).

Hence, we have a(L)⊆Lij. In particular, the linear forms xi=xi◦aand yj◦a

both vanish on L. The latter is equal to yjfor j<dand otherwise it is equal to ℓ. In

either case however, the two forms are linearly independent and therefore cut out the

space L.

111

The action of GQpermutes these linear spaces. Dually, its action on W∗per-

mutes the spaces ⟨xi,yj◦a⟩ ⊆ W∗. Reminiscent of the proof of Theorem 9.1.1, we

can conclude in the same fashion that the action of GQmaps every element of the set

M:={x1, . . . , xd,y1, . . . , yd−1,ℓ}to a scalar multiple of an element of M. Of course, it

can map ydto any linear form which is linearly independent from the other coordi-

nates. Hence, there is a group K⊆GL2d−1such that GQadmits a matrix representa-

tion

GQ=

⎧

⎪

⎨

⎪

⎩

⎛

⎜

⎝

∗ ··· ∗ α

⎞

⎟

⎠

⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐

h∈K,

α∈C×

⎫

⎪

⎬

⎪

⎭

∼

=K×C2d−1×C×. (4)

Claim. K is a finite group of order|K|=r!·d2·(d−1)!.

This claim implies the lemma as follows: We have dim(GQ) = 2d=dim(Hd) + 2,

the latter by Theorem 9.1.1. From Theorem A.1.9, we conclude that ΩQhas codimen-

sion two in Ω(bnd). If Qd−1,d−1and Qd,d−1were in the same orbit, then their stabi-

lizers would be conjugate and in particular, they would both have the same number

of connected components. However, the number of connected components of GQis

equal to|K|, which depends on r.

We are therefore left to verify our claim. We first study the subgroup

Π:={π∈K⏐⏐ℓ◦π∈C×ℓ}

of Kand observe that its order depends on r:

Claim. We have Π∼

=Zdo(Sr×Sd−1), where Zdcorresponds to scaling all variables

with a d-th root of unity simultaneously and Sr×Sd−1permutes the x-variables and

the y-variables that occur in ℓamong themselves.

Proof of Claim. We first check that every π∈Πis a product of a diagonal and a

permutation matrix. Let M:={Cx1, . . . , Cxd,Cy1, . . . , Cyd−1,Cℓ}be the set of lines

spanned by elements of M. We know that πinduces a permtutation of Mwhich

has Cℓas a fixpoint. In particular, πinduces a permutation of the lines spanned by

the first 2d−1 coordinates. This is precisely what we claimed.

Next, we show that all diagonal matrices in Πare scalar and correspond to mul-

tiplication with a d-th root of unity. Let π∈Πbe a diagonal matrix and let ξ∈C×

be such that ℓ◦π=ξℓ. It is straightforward to see that this implies xi◦π=ξxiand

yj◦π=ξyjfor all 1 ≤i,j≤d−1. As πstabilizes

Q=x1···xd+y1···yd−1·(x1+···+xr+y1+···+yd−1),

112

it must, for instance, stabilize the monomial y2

1·y2···yd, whence ξd=1. Let ϑ∈C×

be such that xd◦π=ϑxd. As πalso stabilizes the monomial x1···xd, we also get

ϑξd−1=1, hence ϑ=ξ.

We now assume that π∈Πis a permutation and show that πdoes not map

any x-variable to a y-variable. Observe that any monomial of Qthat is divisible by

ay-variable is divisible by d−1 of the y-variables. Assume for contradiction that

some x-variable is mapped to a y-variable. Then, since (x1···xd)◦π=∏d

i=1(xi◦π)

is a monomial of Q, we conclude that d−1 of the d x-variables must be mapped to

y-variables. In turn, all y-variables must be mapped to x-variables. As Qcontains the

monomial y2

1·y2···yd−1, this implies the contradiction that Q=Q◦πalso contains

the monomial x2

1·x2···xd−1.

We are left to verify that for r=d−1, every permutation π∈Πleaves xdinvariant.

Assume for contradiction that there is some i≤d−1 such that xi◦π=xd. Then, Q=

Q◦πcontains the monomial (y1···yd−1·xi)◦π=y1···yd−1·xd, a contradiction.

As|Π|=r!·d·(d−1)!, we are left to verify that K/Πcontains dresidue classes,

by Lagrange’s Theorem. For any 1 ≤j<d, denote by gj∈Kthe map that satis-

fies yj◦gj=−ℓand is the identity on all other variables. Hence, ℓ◦gj=−yjand it

is easy to check that gjstabilizes Q. We will show that

K/Π={Π,g1Π, . . . , gd−1Π}. (5)

To this end, let g∈Kbe such that ℓ◦gis not a scalar multiple of ℓ. Then, there is

some β∈C×with ℓ◦g=β·z, where zis a variable. Furthermore, there is an η∈C×

such that gmaps a variable ˜

z∈Mto η·ℓ. By applying an element of the subgroup Π,

we may assume that z=˜

z, i.e., we have z◦g=η·ℓ. We may furthermore assume

that gmaps all variables other than zto scalar multiples of themselves. By studying

the monomials of Q, it is straightforward to check that z=yjfor some 1 ≤j<d.

There are certain scalars αiand βksuch that

β·yj=ℓ◦g=(∑r

i=1xi+∑d−1

k=1yk)◦g=(∑r

i=1αixi)+(∑k=jβkyk)+η·ℓ.

With βj:=−β, we obtain that −η·ℓ=∑r

i=1αixi+∑d−1

k=1βkyk, so αi=−ηand βk=−η

for all iand k. In particular, η=β. It is now again easy to check that −ηis a d-th root

of unity, so we have achieved g∈gjKand proved (5).

It is natural to ask what a generic elements of B:=bnd◦S looks like. It turns out

that Bis not an orbit closure. Instead, for the one-parameter family of polynomials

Qt:=t·x1···xd+y1···yd−1·(d

∑

k=1

xk+

d−1

∑

k=1

yk),(t∈C×)

we prove the following result.

113

9.2.4 Proposition. Let B:=bnd◦S. The union U:=⋃t∈C×Ω(ˆ

Qt)is dense in B

and contains an open subset of B. In particular, a generic element of Bis linearly

equivalent to ˆ

Qtfor some t∈C×.

For the proof, we require the following technical lemma whose proof we postpone

until after the proof of Proposition 9.2.4.

9.2.5 Lemma. Let a∈ S with dim(ker(a)) = 1 and P:=bnd◦a. Then, one of the

following is true:

(1) There is some t∈C×such that P∼ˆ

Qt.

(2) There are natural numbers r≤dand s≤d−1 such that P∼Qr,s.

Furthermore, the codimension of ΩPin Bis at least 1, so it is a proper closed subset

of B.

Proof of Proposition 9.2.4.The map ω:S → B,a↦→ bnd◦ais a dominant morphism.

The set S1:={a∈ S |dim(ker(a)) = 1}is open and dense in S. Since ωis dominant,

this means that ω(S1)is dense and contains a nonempty open subset, the latter by

[TY05, 15.4.3]. By Lemma 9.2.5, we have ω(S1)\Z=U, where

Z:=⋃d

r=1⋃d−1

s=1Ω(Qr,s).

By Lemma 9.2.5,Zis a finite union of proper closed subsets of B, therefore Zitself is

a proper closed subset of B. Consequently, Uis dense and contains a nonempty open

subset.

9.2.6 Remark. Of course, it is equally true that the set

⋃

t∈C×

Ω(ˆ

Qt)∪⋃

1≤r≤d

1≤s<d

Ω(Qr,s)

is dense in Band contains an open subset, the statement of Proposition 9.2.4 merely

emphasizes the fact that the Qr,sare just a finite number of “special” cases.

Proof of Lemma 9.2.5.By Lemma 8.3.3, we have dim(GP)≥2dand by Theorem 9.1.1,

this means dim(GP) = dim(Hd) + 2. This implies that ΩPhas codimension at least 2

in Ω(bnd)by Theorem A.1.9. Proposition 9.2.2 then yields that ΩPhas codimension

at least 1 in B.

We will show later that there is a binary vector ε∈{0,1}2d−1and some t∈C×

such that P=bnd◦ais linearly equivalent to the polynomial

Pε,t:=t·x1···xd+y1···yd−1·(d

∑

k=1

εkxk+

d−1

∑

k=1

εd+kyk)(6)

114

We check first that Pε,tis linearly equivalent to some ˆ

Qtor to some Qr,s. Let rbe the

number of indices 1 ≤k≤dwith εk=1 and let sthe number of indices 1 ≤k<d

with εd+k=1. By permutation of the variables we get

P∼Pε,t∼t·x1···xd+y1···yd−1·(r

∑

k=1

xk+

∑

k=1

yk).

If r=dand s=d−1, the above means P∼ˆ

Qt. If we have r<d, then we can scale

xdby t−1to achieve P∼Qr,s. The final case is s<d−1. In this case, choose ϑ∈Cbe

so that ϑd=t. Scaling yd−1by ϑ1−dand all other variables by ϑalso yields P∼Qr,s.

We will compute the codimension of Ω(Pε,t)last and first show that P∼Pε,t. We

assume d=3, making the proof easier to read and follow. It generalizes easily to

general d. We will show that there are h∈Hdand g∈GL(W)such that

h◦a◦g=⎛

⎜

⎝

t0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

ε1ε2ε3ε4ε50

⎞

⎟

⎠(7)

for a certain binary vector εand t∈C×. Then,

Pε,t=bnd◦(h◦a◦g) = bnd◦(a◦g) = P◦g.

To verify (7), we will apply elements of Hdfrom the left and perform arbitrary column

operations to transform ainto the right hand side of (7). By column operations, we

can achieve

a=⎛

⎝

∗0 0 0 0 0

0∗0 0 0 0

0 0 ∗0 0 0

000∗0 0

0 0 0 0 ∗0

∗∗∗∗∗0

⎞

⎠

with all diagonal entries nonzero. Composing from the right with another element

of GL(W), we may scale all columns to achieve that

a=⎛

⎝

∗0 0 0 0 0

0∗0 0 0 0

0 0 ∗0 0 0

0 0 0 ∗0 0

0 0 0 0 ∗0

ε1ε2ε3ε4ε50⎞

⎠

for some binary vector ε. After precomposing awith an element of Hd, we may

assume that there are ˜

η,η∈C×with

a=⎛

⎜

⎝

η0 0 0 0 0

0˜

η0 0 0 0

0 0 ˜

η0 0 0

0 0 0 η0 0

0 0 0 0 η0

ε1η ε2η ε3η ε4η ε5η0

⎞

⎟

⎠.

115

We now scale every column with η−1and set ϑ:=˜

ηη−1, obtaining the form

a=⎛

⎜

⎝

ϑ0 0 0 0 0

0ϑ0 0 0 0

0 0 ϑ0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

ε1ε2ε3ε4ε50

⎞

⎟

⎠

Finally, set t:=ϑd. We multiply by diag(ϑd−1,ϑ−1, . . . , ϑ−1, 1, . . . , 1)∈Hdfrom the

left to achieve

a=⎛

⎜

⎝

t0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

ε1ε2ε3ε4ε50

⎞

⎟

⎠.

9.3 The Second Boundary Component

We will use the linear subspace L0to construct a concise degeneration Qof bnd. We

choose the linear map b:W→L0as in (2), i.e., we have xi◦b=xiand yi◦b=ζxifor

all 1 ≤i≤d, where ζd=−1. Consider the linear approximation

bnd◦(b+t·idW) = (t+1)d(x1···xd) +

∏

i=1

(ζxi+tyi).

The coefficient of tin this expression is the polynomial

Q:=

∑

k=1(yk·ζd−1∏d

ℓ=1

ℓ=kxℓ)+d·

∏

ℓ=1

xℓ.

Scaling each xℓby d

√d−1and each ykby ζd+1d

√dd−1, we get the polynomial

Q:=

∑

k=1(yk·∏d

ℓ=1

ℓ=kxℓ)+

∏

ℓ=1

xℓ,

which satisfies Q∈˜

Q◦GL(W)⊆Ω(bnd).

We now prove that Qis concise and compute the dimension of its stabilizer:

9.3.1 Proposition. The polynomial Qis concise and dim(G◦

Q) = 2d−1.

Furthermore, the identity component G◦

Qof GQconsists of all matrices

(s0

u t )=

⎛

⎜

⎝

s10

...0

0sd

u10t10

......

0ud0td

⎞

⎟

⎠

(8)

such that tj=∏i=js−1

ifor all 1 ≤j≤dand 1 =∏d

i=1si+∑d

j=1uj∏i=jsi.

116

Proof. We will compute the Lie algebra of G◦

Q. In the process, we will also compute

the partial derivatives of Qand note in passing that Qis concise.

Denote by Y⊆GL(W)the closed subvariety of all matrices of the form (8), subject

to the listed conditions. We use coordinates (x1, . . . , xd,y1, . . . , yd)on the space W=

Cd×Cd. In other words, xiis mapped to sixiand yjis mapped to ujxj+tjyj. It is a

straightforward computation to verify that any element of Ystabilizes Q, so Y⊆G◦

This implies that we have dim(G◦

Q)≥dim(Y)≥3d−(d+1) = 2d−1 since any of

the given (d+1)equations can reduce the dimension by at most one.

We now show that the dimension of the Lie algebra Lie(GQ)is at most 2d−1. This

gives us

2d−1≤dim(GQ) = dim(Lie(GQ)) ≤2d−1,

therefore we have equality. It follows in particular that Yis an irreducible component

of GQcontaining the identity, so Y=G◦

Recall the action of gl(W)on V=C[W]dfrom the proof of Theorem 9.1.1. We

again consider the elements of gl(W)as block matrices (s v

u t )where s,u,v,t∈Cd×d.

Such a block matrix is in Lie(GQ)if and only if

∑

i=1

∑

j=1(sij ·(xi∂Q

∂xj)+uij ·(xi∂Q

∂yj)+vij ·(yi∂Q

∂xj)+tij ·(yi∂Q

∂yj)). (9)

We set µj:=x1···xd

xj=∂Q

∂yjand similarly µjk :=x1···xd

xjxk. Then, we have

∂Q

∂xj=

∑

k=1

k=j

⎛

⎜

⎝yk·

∏

ℓ=1

ℓ=j,ℓ=k

xℓ⎞

⎟

⎠+

∏

ℓ=1

ℓ=j

xℓ=

∑

k=1

k=j

ykµjk +µj.

We note that the partial derivatives of Qare all linearly independent, hence Qis

concise – this proves part of the statement. Plugging this into (9) yields

∑

i=1

∑

j=1(uijxiµj+tijyiµj+(∑

k=j

sijxiykµjk)+sijxiµj+(∑

k=j

vijyiykµjk)+vijyiµj)

∑

i=1

∑

j=1((sij +uij)xiµj+ (tij +vij)yiµj+(∑

k=j

sijxiykµjk)+(∑

k=j

vijyiykµjk)).

(10)

We have summarized some coefficients of certain monomials appearing in the

right hand side of (10) in Figure 9.3.1. Coefficient 6implies that vis the zero matrix

and Coefficient 5implies that s=diag(s1, . . . , sd)is diagonal. Note that we can pick

any k/∈{i,j}for Coefficients 5and 6, and we require the condition d≥3 for such a k

117

No. Monomial Condition Coefficient

(1) xiµji=j sij +uij

(2) xjµj∑d

i=1sii +∑d

i=1uii

(3) yiµji=j tij +vij +sij

(4) ykµktkk +vkk +∑j=ksjj

(5) xiykµjk i=j,k/∈{i,j}sij

(6) yiykµjk j=k vij

Figure 9.3.1: Coefficients of certain monomials occuring in (10)

to exist. Coefficient 1therefore means that u=diag(u1, . . . , ud)is also diagonal, and

Coefficient 2becomes d

∑

i=1

ui+

∑

i=1

si=0. (11)

Finally, Coefficient 3implies that t=diag(t1, . . . , td)is also diagonal and Coefficient 4

means

tk=−∑d

j=1

j=k

sj.

Hence, the only parameters that remain are the diagonal entries of sand u, modulo

the relation (11). This proves dim(Lie(GQ)) ≤2d−1 as required.

9.3.2 Corollary. The orbit closure Ω(Q)is a component of ∂Ω(bnd)which is not con-

tained in bnd◦End(W).

Proof. Proposition 9.3.1 implies that Ω(Q)is an irreducible closed subvariety of the

boundary ∂Ω(bnd). It is not contained in bnd◦End(W)because Qis concise. Fur-

thermore, this variety has dimension dim(Ω(Q)) = 4d2−2d+1=dim(Ω(bnd)) −1,

by Corollary 9.1.2 and Theorem A.1.9.

9.4 The Indeterminacy Locus

We will ommit bndas a subscript in this chapter to ease the notation, i.e., we denote

by Athe annihilator of bndand by ˆ

Athe subscheme of End(W)given by the equations

bnd◦a=0 in the coordinates a.

We have constructed one concise component of the boundary, but we do not know

if there are more, potentially depending on d. Unfortunately, we cannot determine

the complete scheme structure of ˆ

Ass. We know that Ass =Hd◦End(W,L0)ss by (3).

We will see that Ass has d! irreducible components and each of these components is

smooth. However, the scheme ˆ

Ass will be singular where these components intersect.

118

9.4.1 Remark. If ˆ

Ass is reduced and its irreducible components intersect transver-

sally, the blowup β:Γ→End(W)ss with center ˆ

Ass =Ass is a sequence of smooth

blowups, one for each component [Li09, Thm. 1.3]. Moreover, in this case the stabi-

lizer of the binomial acts transitively on the irreducible components of β−1(Ass)and

by Remark 7.3.12, it follows that Ω(Q)is the only concise component of ∂Ω(bnd).

We emphasize that this remark is not to be mistaken as a conjecture – it is quite

unclear whether or not ˆ

Ass is even reduced at the intersection of its components. That

said, we can still show the following:

9.4.2 Theorem. For any point y∈ Ass that lies outside the intersection of two ir-

reducible components of Ass, the scheme ˆ

Ass is smooth at y. In particular, ˆ

Ass is

generically smooth and in particular generically reduced.

The remainder of this section is dedicated to the proof of Theorem 9.4.2.

9.4.3. Consider b=(I0

ζI0), then Ass = (Hd◦b◦GL(W))ss. Let T⊆SLdbe the group

of diagonal matrices with determinant one. We define

Y1:=H◦

d◦b◦GL(W) ={(s0

0t)( I0

ζI0)(g1g2

g3g4)⏐⏐(s0

0t)∈H◦

d,(g1g2

g3g4)∈GL(W)}

={(sa

ζa)⏐⏐⏐s∈Tand a∈Cd×2d}.

For a permutation π∈Sd, which we understand as a d×dpermutation matrix, we

furthermore define

Yπ:=(π0

0I)◦Y1={(πsa

ζa)⏐⏐⏐s∈Tand a∈Cd×2d}.

9.4.4 Proposition. The irreducible components of Y:=Hd◦b◦GL(W)are the Yπfor

all permutations π∈Sdand these components are pairwise distinct.

In particular, Yhas d! irreducible components. A point (πsa

ζa)∈Yis semistable if

and only if all rows of aare nonzero.

Proof. Recall the permutation group K⊆GL(W)from Theorem 9.1.1 which permutes

the first and last dcoordinates among themselves and which also swaps the xand y

coordinates simultaneously. Since Hd=K·H◦

d, we have Y=K·Y1.

We first note that Y1is invariant under the permutation switching the xand y

coordinates simultaneously: For s∈Tand a∈Cd×2d, set t:=ζ2s−1∈Tand c:=

ζ−1sa. Then, (ζa

sa )=(tc

ζc).

Furthermore, we will see that permutations on the first dcoordinates suffice: For

ϱ,σ∈Sdlet π:=ϱσ−1and b:=σato see that

(ϱ0

0σ)(sa

ζa)=(ϱsa

σζa)=(πsb

ζb).

119

Assume now that Yπ=Yσfor two permutations πand σ. We have (π

ζI)∈Yπ=Yσ

and hence, there are a∈Cd×2dand an s∈Twith (π

ζI)=(σsa

ζa). This implies that we

have a=Iand hence, σs=π, so s=σ−1π. However since sis diagonal, this means

it must be the identity and therefore σ=π. This proves that the Yπare the pairwise

distinct irreducible components of Y.

The semistability condition follows immediately from the description of the maxi-

mal linear subspaces in Section 9.1.

9.4.5 Corollary. A point (πsa

ζa)∈Yπlies in the intersection of Yπwith another com-

ponent Yσof Yif and only if there are two rows of a∈Cd×2dthat are scalar multiples

of one another.

Proof. If the i-th and j-th row of aare scalar multiples of one another, let σbe the

composition of πwith the transposition τ:= (i,j). By assumption, there is a t∈T

with tτa=sa, so we have πsa =πτtτa=σtτa=σsa. Hence, a∈Yπ∩Yσ.

Conversely, assume that there is a permutation σ=πwith (πsa

ζa)=(σtc

ζc)for

some t∈Tand c∈Cd×2d. It follows immediately that c=a, hence πsa =σta.

Denoting by aithe i-th row of a, this means sπ(i)aπ(i)=tσ(i)aσ(i)for all 1 ≤i≤d. As

π=σ, there is an index iwith k:=π(i)=σ(i) =:j, so we have skak=tjaj, therefore

ak=s−1

ktjaj.

The proof of Theorem 9.4.2 is now completed by the following proposition.

9.4.6 Proposition. Let a∈Cd×2dbe such that no row of ais a scalar multiple of

another. Let also π∈Sdand u∈T. Then, y:=(πua

ζa)is a smooth point of ˆ

Ass.

Proof. We first note that dim(Y) = dim(Y1) = 2d2+d−1, hence we have to show that

the tangent space Tyˆ

Ass has the same dimension.

As Hd×GL(W)and Sdact by automorphisms on ˆ

Ass, the tangent space at yis

isomorphic to the tangent space of any element in the Hd×GL(W)orbit of y. Hence,

we may assume y=(a

ζa).

Note that each row of ais nonzero. After applying some column operation

from GL(W)we may assume that ai1=0 and the last dcolumns of aare zero.

Scaling the first column by the inverse of a11 ···ad1with another column operation,

we achieve that a11 ···ad1=1. We then apply a row scaling operation from Hdto

achieve ai1=1 for all 1 ≤i≤d. We set ai:=xi◦a=x1+∑d

j=2aijxj. Since the aiare

not scalar multiples of one another, we achieve ai=x1+cix2+˜

aiwith further column

operations, where the ciare all distinct and the ˜

aiare linear forms in x3, . . . , xd.

120

Let b=(s v

u t )∈TyEnd(W) = End(W)be a tangent vector, where s,v,u,t∈Cd×d.

The polynomial bnd◦(a+bT)is equal to

∏

i=1(ai+

∑

j=1

sijxjT+

∑

j=1

vijyjT)+

∏

i=1(ζai+

∑

j=1

uijxjT+

∑

j=1

tijyjT).

Set fi:=∏k=iak, then the coefficient of Tin this expression is equal to

∑

i=1

∑

j=1

sij fixj+vij fiyj−ζ−1d

∑

i=1

∑

j=1

uij fixj+tij fiyj

∑

i=1

∑

j=1(sij −ζ−1uij)fixj+(vij −ζ−1tij)fiyj. (12)

By the Jacobian criterion, b∈Tyˆ

Ass is equivalent to the vanishing of (12) as a poly-

nomial in the xjand the yj. We first note that the fi∈C[x1, . . . , xd]d−1are linearly

independent polynomials: Indeed,

fi=˜

fi+∏

k=i

(x1+ckx2+˜

ak) = ∏

k=i

(x1+ckx2)(13)

for certain polynomials ˜

fi∈C[x1, . . . , xd]d−1whose monomials all contain some vari-

able xjwith j>2. Hence, it is sufficient to show that the gi:=∏k=i(x1+ckx2)are

linearly independent. Assume that 0 =∑d

i=1λigifor certain λi∈C. For any k, we can

evaluate this expression at the point (−ck,1)and since gkis the only function from

among the githat does not vanish, it follows that λk=0.

Since we have established that the fiare linearly independent and do not use the

y-variables, it follows immediately that the coefficients of fiyjin (12) must all vanish,

i.e., we have v=ζt. This constitutes d2linear conditions.

If we can show that the polynomials fixjspan a subspace of dimension at least

d2−d+1 inside C[x1, . . . , xd]d, we are done: It gives us at least 2d2−d+1 linear

conditions on the parameters (s v

u t )and hence,

dimTyˆ

Ass ≤4d2−(2d2−d+1) = 2d2+d−1=dim(Y).

Since the other inequality always holds, this implies equality. Now, recall from (13)

that fi=˜

fi+gi. For 1 ≤k≤d, let Ukbe the span of all monomials that have combined

degree kin{x1,x2}, so we have C[x1, . . . , xd]d=⨁d

k=0Uk. We have fixj∈⨁d−1

k=0Ukfor

all 2 <j≤dand the projection to Ud−1is equal to gixj. These polynomials span a

space of dimension d2−2dbecause the giare linearly independent and do not contain

the variables xjwhen 2 <j≤d.

For j∈{1,2}, the projection of fixjto Udis equal to gixjand it will suffice to show

that these polynomials span a space of dimension at least d+1. Write these vectors in

121

coordinates with respect to the basis (x0

1xd

2,x1

1xd−1

2,x2

1xd−2

2, . . . , xd

1x0

2), then they form

the columns of a (d+1)×(2d)matrix

M:=

g1x1··· gdx1g1x2··· gdx2

⎛

⎜

⎝

⎞

⎟

⎠

0··· 0

with g∈GLdbecause the giare linearly independent. We have

M·(g−10

0g−1)=⎛

⎜

⎝

⎞

⎟

⎠

∗ ··· ∗ 1

1...

...1

1∗ ··· ∗

which is easily seen to have full rank.

As explained in Remark 9.4.1, it would be interesting to know whether ˆ

Ass is

reduced everywhere:

9.4.7 Question.

(1) Is the scheme ˆ

Ass

bndCohen-Macaulay?

(2) Is the scheme ˆ

Ass

bndreduced?

Remark. Note that (1) implies (2) by Theorem 9.4.2 and the fact that Cohen-Macaulay

schemes have no embedded components [GW10, Prop. 14.124].

122

Part III

Appendix

123

Appendix A

Algebraic Groups and Representation Theory

Throughout this chapter, we fix an algebraically closed field Cof characteristic zero.

We assume knowledge of classical algebraic geometry, see [Har06;Har95] and also

[Kra85, Anhang]. We denote the coordinate ring of an affine C-variety Xby C[X]. We

also use the notation [n]:={1, . . . , n}.

A.1 Algebraic Semigroups and Groups

If Eis a semigroup acting on a set X, we usually denote this action by a dot, so

the action of g∈Eon x∈Xis denoted g.x. The set Ex:={g∈E|g.x=x}is the

stabilizer of x∈X. The orbit of xunder the action of Eis the set E.x:={g.x|g∈E}.

If E.x={x}, we say that xis E-invariant. If S⊆Eis any subset, we denote by

XS:={x∈X|∀s∈S:s.x=x}the set of all S-invariant elements of X. A subset

Y⊆Xis E-stable if E.Y⊆Y. If Eis a monoid, we write 1or 1Efor the unique neutral

element of E.

A.1.1 Definition. An algebraic semigroup is a semigroup (E,◦)which is also a C-

variety such that the composition map

µE:E×E−→ E

(g,h)↦−→ g◦h

is a morphism of varieties. A morphism of algebraic semigroups is a morphism

φ:E→E′of varieties between two algebraic semigroups which is also a morphism

of semigroups. An left-E-variety is a variety Xtogether with a morphism

αX:E×X−→ X

which defines a left E-action on the set X. Similarly, the notion of a right-E-variety is

defined. An E-variety is defined to be a left-E-variety. A morphism of E-varieties is

a map φ:X→Ywhich is a morphism of varieties such that φ(g.x) = g.φ(x)for all

g∈Eand x∈X.

125

A.1.2 Definition. An algebraic group (monoid) is an algebraic semigroup Ewhich is

also a group (monoid). The unit group of an algebraic monoid Eis defined as the

group G(E):={g∈E|1∈(g◦E)∩(E◦g)}.

A.1.3 Proposition. The unit group of an algebraic monoid is an algebraic group. If

Gis an algebraic group, then Gis a smooth variety and the inversion morphism

ιG:G→Gmapping g↦→ g−1is a morphism of varieties.

Proof. We first show that Gis smooth. The set of regular points of Gis open and

dense, therefore not empty. Let’s assume that Gis regular at g. Given any other point

h∈G, since the multiplication with the group element hg−1is an automorphism of

G, it follows that h=hg−1gis also a nonsingular point.

Let Ebe an algebraic monoid and G=G(E). The set Γ:=µ−1

E(1)⊆E×Eis

a closed subvariety because it is the preimage of a closed point. For i∈{1,2}we

denote by pri:E×E→Ethe projections. Then, G=pr1(Γ)∩pr2(Γ)is constructible

by [TY05, 15.4.3]. This means that Gis locally closed. Hence, it contains an open

subset Uof its closure Gin E. Because Gacts transitively on itself, we see that

G=⋃g∈Gg.Uis open in G, therefore a variety and consequently, an algebraic group.

Since any algebraic group Gis the unit group of itself, we can maintain the above

notation for the rest of the proof. We now show that ιGis a morphism of varieties.

Note that Γis the graph of the map ιG. Let πi:=pri|Γbe the restrictions of the

projection morphisms to Γ. Each of the πiis a bijective map from a variety to a smooth

variety, so by [TY05, Corollary 17.4.8] they are both isomorphisms of varieties. This

implies that ιG=π−1

2◦π1is a morphism.

A.1.4 Example. Let W∼

=Cnand E=End(W)∼

=Cn×n.Eis an algebraic monoid

and the unit group of Eis the general linear group G=GL(W)of invertible linear

maps W→W. By choosing a basis of W, we can identify Gwith the group GLnof

invertible complex n×nmatrices. We consider the set V=C[W]dof homogeneous d-

forms on W. It is a finite-dimensional C-vector space, hence a variety. We define the

(right) action αV:V×E→Vvia αV(P,a):=P◦a.

A.1.5 Remark. Let Gbe an algebraic group. If Xis an affine G-variety, we have an

induced G-action on the coordinate ring C[X]by g.ϕ:= (g−1)∗(ϕ) = ϕ◦g−1, where

we understand g−1:X→Xas a morphism of varieties. Indeed, for g,h∈Gand

ϕ∈C[X],g.h.ϕ= (g−1)∗((h−1)∗(ϕ)) = (h−1g−1)∗(ϕ) = ((gh)−1)∗(ϕ) = gh.ϕ.

The inverse is required to obtain a left action rather than a right action.

The following result will be important to reduce to the connected case:

126

A.1.6 Proposition. Let Gbe an algebraic monoid. There is a unique irreducible com-

ponent G◦⊆Gcontaining 1.G◦is a closed submonoid of G.

If Gis an algebraic group, then G◦is a normal subgroup and the irreducible

components of Gare the cosets hG◦for h∈G. In particular, the quotient group G/G◦

is finite.

Proof. Let G◦be a connected component of Gcontaining 1and let Xbe any irreducible

component of G. Then XG◦is the image of the restriction of µGto X×G◦. This means

that XG◦is the image of an irreducible variety under a regular map, therefore XG◦is

an irreducible, closed subvariety of G. Since 1∈G◦, we have X⊆XG◦⊆XG◦.Xis

irreducible, so X=XG◦. This implies X=XG◦and similarly, we obtain X=G◦X.

In particular, G◦=G◦G◦, so G◦is a closed submonoid. If 1was also contained in X,

then we could similarly conclude that G◦X=XG◦=G◦, so we have X=G◦.

Assume now that Gis a group. If Xis an irreducible component, then for any

h∈X, we know that h−1Xis the image of an irreducible set under the automorphism

h, so it is an irreducible component. Since 1∈h−1X, we know that h−1X=G◦, hence

X=hG◦. For any h∈G, similarly observe that hG◦h−1is a component containing 1,

so hG◦h−1=G◦. This proves that G◦is a normal subgroup. Since its cosets are the

(finitely many) irreducible components of X, this proves the statement.

A.1.1 Quotients by Algebraic Groups

Let Xbe a variety on which an algebraic group Gacts. Then, a categorical quotient

is a morphism π:X→Ysuch that

(C1) πis constant on G-orbits.

(C2) If ψ:X→Zis any morphism of varieties which is constant on G-orbits, then

there exists a unique ¯

ψ:Y→Zwith ¯

ψ◦π=ψ.

If a categorical quotient exists, it is unique up to unique isomorphism and we

denote it by Y=X//G.

However, categorical quotients can lack many properties one would expect from

the quotient by a group action. For example, there are cases when the projection

morphism is not surjective. A much stronger notion is the one of a good quotient,

defined to be a morphism π:X→Ywith the following properties:

(G1) The morphism πis G-invariant and surjective.

(G2) The comorphism of πinduces an isomorphism OY∼

=(π∗OX)G.

(G3) For a closed and G-invariant subset Z⊆X, the image π(Z)is closed in Y.

(G4) Two closed, G-invariant subsets Z1,Z2⊆Xare disjoint if and only if π(Z1)

and π(Z2)are disjoint.

127

A good quotient is called a geometric quotient if the fibers of πare precisely the

G-orbits of X.

A.1.7 Remark. See [TY05, 25.2] for some remarks on Property (G2). The gist is that

for any open subset U⊆Y, the ring (π∗OX)(U) = OX(π−1(U)) carries a G-module

structure, so we can consider the corresponding subring of G-invariant functions

OX(π−1(U))G. This defines a subsheaf of rings (π∗OX)G⊆(π∗OX).

By [New78, §4, Prop. 3.11], any good quotient is also a categorical quotient. If the

quotient is geometric, we denote it by X/G.

A.1.8 Theorem. Let Gbe an affine, reductive algebraic group acting on an affine vari-

ety X. The ring C[X]Gis a finitely generated C-algebra and Y=Spec(C[X]G)together

with the surjective morphism π:X→Yinduced by C[X]G⊆C[X]defines a good

quotient of Xby G. Furthermore:

(1) If Gis finite and Xirreducible, then this quotient is geometric.

(2) For all x∈X, the fiber π−1(π(x)) contains a unique closed G-orbit Ω. Further-

more, we have π−1(π(x)) ={y∈X⏐⏐Ω⊆G.y}.

Proof. See [TY05, 27.5] and [TY05, 25.5.2].

If a group Gacts on a set X, one can identify the orbit G.xof any element x∈X

with the set of cosets G/Gxwhere Gx={g∈G|g.x=x}is the stabilizer of x. It

turns out that under certain assumptions, this remains valid and compatible when X

is a G-variety.

A.1.9 Theorem ([TY05, 21.4]). Let Gbe an affine algebraic group and XaG-variety.

For every point x∈X, we have:

(1) The orbit G.xis a smooth subvariety of Xwhich is open in its closure G.x.

(2) The stabilizer Gxis a closed subgroup of Gand every component of G.xhas

dimension dim(G)−dim(Gx).

(3) G.xis the union of G.xand orbits of strictly smaller dimension.

(4) If Xis affine and Gxis reductive, the right action of Gxon Gby multiplication

admits the good quotient G.x∼

=G//Gxand in particular G.xis an affine variety

with coordinate ring C[G.x]∼

=C[G]Gx.

A.1.2 Nullcone and Projective Quotients

If an algebraic group Gacts on a vector space V, the set N:={v∈V⏐⏐0∈G.v}

is called the nullcone of this action. Especially when Gis reductive, the geometry

128

of Ncarries a lot of information about the quotient, see [Kra85]. We will require the

following observations:

A.1.10 Lemma. Let Gbe a reductive affine algebraic group acting on a complex vector

space V. Let π:VV//Gbe the quotient and N ⊆ Vthe nullcone. Then,

•N=π−1(π(0)), in particular Nis a closed subset of V.

•N={v∈V⏐⏐0∈G◦.v}where G◦is the connected identity component of G.

•Nis the vanishing set of all homogeneous, G-invariant functions in C[V].

Proof. Since the origin 0 is a closed G-orbit, we have 0 ∈G.vif and only if π(v) = π(0)

according to Theorem A.1.8.

For the second item, note that G=H·G◦where His a finite set, according to

Proposition A.1.6. Since G.v=H.G◦.v=H.G◦.v, we can see that 0 ∈G.vif and only

if 0 ∈G◦.v.

Finally, the defining ideal of π(0)in C[V]Gis equal to the maximal ideal generated

by all homogeneous elements of positive degree, and since πcorresponds to the in-

clusion C[V]G⊆C[V], the set Nis the vanishing set of the same elements, considered

as functions on V.

If Vis a G-module, the projective space P(V)is a G-variety in a natural way. A

linearized projective G-variety is a projective variety Xwith a fixed embedding as

a closed, G-invariant subset X⊆P(V)for some G-module V. We write ˇ

X⊆V

for the affine cone over X, i.e., ˇ

Xis an affine variety which is closed under scalar

multiplication and X=P(ˇ

X).

If Xis a linearized projective G-invariant closed cone, then C[ˇ

X]is a graded alge-

bra. It is the projective coordinate ring of X. The invariant algebra C[ˇ

X]Ginherits this

grading and in particular, one can consider the projective variety Y:=Proj(C[ˇ

X]G)

and the induced rational map π:X99K Y. If we denote by [x]∈Xthe projective

class of a point x∈ˇ

X, then π([x]) = [ ˇ

π(x)] where ˇ

π:ˇ

Xˇ

X//Gis the affine quo-

tient from Theorem A.1.8. This rational map is undefined when ˇ

π(x) = 0, which by

Lemma A.1.10 means that PNis the indeterminacy locus of π. Hence, Xss :=X\PN

is the open set where πis defined. It is called the set of semistable points of X, with

respect to the (induced) action of Gon X. One also calls ˇ

Xss :=ˇ

X\N the set of

semistable points of ˇ

X. Since Nis a cone, we have Xss = (Pˇ

X)ss =P(ˇ

Xss)and we

sometimes omit the brackets. The restriction π:Xss →Yis a morphism.

A.1.11 Proposition. Let Xbe a linearized projective G-variety and Sits projective

coordinate ring. The projective variety Y:=Proj(SG)together with the morphism

π:Xss →Yinduced by SG⊆Sdefines a good quotient Y=Xss//G.

129

Using Theorem A.1.8, the proof is straightforward. We omit it for brevity and refer

to [Dol03, Thm. 8.1, Prop. 8.1] for a modern treatment and [New78, §4, Thm. 3.14] for

a more classical version. We note the following amplification of Property (C2):

A.1.12 Proposition. Let Xbe a linearized projective G-variety and let ϖ:X99K Ybe

aG-invariant rational map between projective varieties. Then, there exists a rational

map φ:Xss//G99K Zsuch that the following diagram commutes:

Xss

↘↘↘↘

ϖ→→Y

Xss//G

↗↗

Here, πis the quotient morphism.

Proof. Fix some embedding of Yin a projective space. Let Sand Rbe the projective

coordinate rings of Xand Y, respectively. The rational map ϖcorresponds to a homo-

morphism ϖ♯:R→Sof graded C-algebras and the assumption that ϖis G-invariant

means that the image of ϖ♯is contained in the invariant ring SG. Since SGis the

projective coordinate ring of Xss//G, this homomorphism R→SGinduces a rational

map φ:Xss//G99K Y. The inclusion SG↪→Sof graded rings corresponds to the

rational map ˜

π:X99K Xss//Gwhich restricts to π, so we have φ◦˜

π=ϖas rational

maps. Restricting to Xss yields the result.

A.2 Representation Theory of Reductive Groups

Let Gbe an affine algebraic group throughout this section. A Borel subgroup of Gis

a maximal connected solvable subgroup Bof G. It follows from the definition that a

Borel subgroup of Gis closed, because if H⊆Gis connected and solvable, then so is

its closure. By [Hum98, 21.3], G/Bis a projective variety and all Borel subgroups of

Gare conjugate. We fix one Borel Bof G.

The only connected, algebraic C-groups of dimension one are the multiplicative

group Gm:=Gm(C):= (C×,·)and the additive group Ga:=Ga(C):= (C,+), see

[TY05, 22.6.2]. A product of multiplicative groups Gr

mis called a torus.

Gcontains a unique largest normal solvable subgroup [TY05, 27.1.1], which is au-

tomatically closed. Its identity component is then the largest connected normal solv-

able subgroup of Gand is called the radical of G, denoted R(G). By [TY05, 27.1.2],

the set of unipotent elements of R(G)is the largest connected normal unipotent sub-

group of G. It is denoted by Ru(G)and we call it the unipotent radical of G. An

affine algebraic group Gis called semisimple (resp. reductive) if R(G)(resp. Ru(G))

is trivial. Note that we follow [TY05, 27.2] here and do not require Gto be connected.

130

However, an algebraic group Ghas a unique, connected, normal subgroup G◦con-

taining the neutral element [TY05, 21.1.5 & 21.1.6] and by definition, Gis semisimple

(resp. reductive) if and only G◦has this property.

For example, each torus is reductive. Furthermore, GLnis reductive, but not

semisimple. However, the subgroup SLn⊆GLnof matrices with determinant 1 is

semisimple.

By [Hum98, 21.3], the maximal tori and the maximal unipotent subgroups of G

are those of the Borel subgroups of G, and all of them are also conjugate. We fix a

maximal torus T⊆Band denote by U=Ru(B)the unipotent radical of B. We have

B=UoTby [Hum98, 19.3].

A.2.1 Example. The affine space Cn×nof all n×nmatrices with entries in Chas the

coordinate ring C[xij ⏐⏐i,j∈[n]]. The group G:=GLnof invertible n×nmatrices is

the set of matrices where detndoes not vanish and therefore a Zariski-open subset

of Cn×n. Hence, Gis an affine algebraic group with coordinate ring C[xij, det−1

n].

Denote by S:=SLnthe set of matrices with determinant 1, i.e., the vanishing set of

detn−1. It is a closed, affine algebraic subgroup of G.

Denote by Inbe the unit matrix, then it is well-known that the center of G=GLn

is equal to Z:=C×·In. By [Lan02, Part 3, Chapter XIII, 8 & 9], the projective linear

group PGLn=G/Z=S/(S∩Z)is simple. Thus, if REGis solvable, then R/Z

is also solvable and can only be trivial or equal to PGLn. However, PGLnis not

solvable because Sis perfect (i.e.,[S,S]=S) and therefore not solvable. Thus, R/Zis

trivial, which means R⊆Z. Since Zis connected, normal and solvable, we must have

R(G) = Z. The only unipotent central matrix is the unit matrix, so Ru(G)is trivial

and we have verified that Gis reductive.

A maximal Borel subgroup Bof Gis given by the upper triangular matrices of

nonvanishing determinant and its unipotent radical Uis the subgroup of those ele-

ments where all diagonal entries are equal to 1. The maximal torus Tcorresponding

to this choice are the invertible diagonal matrices.

A.2.1 Representations

Let Gbe an algebraic group. A representation of Gis a homomorphism of algebraic

groups ϱ:G→GL(V(ϱ)), where V(ϱ)is some finite-dimensional C-vector space. The

space V(ϱ)becomes a G-variety via the induced action. We also write V

G(ϱ)instead

of V(ϱ)if we want to put emphasis on the group. The degree of ϱis defined to be

dim(V(ϱ)).

Conversely, if Vis some C-vector space which is also a G-variety, we say that Vis

aG-module. We remark that the G-modules are in bijection with the representations

131

of Gby associating to a representation ϱ:G→GL(V)the action α:G×V→Vwhere

α(g,v):=ϱ(g)(v).

Ahomomorphism of G-representations is a C-linear map φ:V

G(ϱ1)→V

G(ϱ2)

such that ϱ2(g)◦φ=φ◦ϱ1(g)for all g∈G. Equivalently, a homomorphism of G-

modules is a C-linear map φ:V1→V2where φ(g.x) = g.φ(x)for all g∈G.

We say that ϱ1is a subrepresentation of ϱ2if there is an injective homomorphism

of G-representations V

G(ϱ1)↪→V

G(ϱ2). Equivalently, a submodule of a G-module

V1is a linear subspace V1⊆V2such that V1is stable under the action of G. A

representation ϱis irreducible if ϱhas no nontrivial subrepresentations. Equivalently

aG-module Vis irreducible if it has no G-invariant linear subspaces other than V

and {0}. Denote by Rep(G)the set of all equivalence classes of G-representations

and by Irr(G)⊆Rep(G)the subset of all irreducible representations. If H⊆Gis a

subgroup, we have a canonical restriction map Rep(G)→Rep(H),ϱ↦→ ϱ|H.

The arguably most important lemma of representation theory is the following:

A.2.2 Lemma (Schur’s Lemma). Let Gbe an algebraic group and ϱ1,ϱ2∈Irr(G). Ev-

ery nonzero homomorphism φ:V(ϱ1)→V(ϱ2)of G-representations is an isomor-

phism.

Proof. Note that for v∈ker(φ), we have φ(g.v) = g.φ(v) = g.0 =0, hence ker(φ)is a

G-invariant subspace of V(ϱ1). Since φis nonzero, we must have ker(φ) ={0}.

For two representations ϱ1,ϱ2∈Rep(G)with Vi:=V(ϱi), we define the repre-

sentation ϱ1⊕ϱ2:G→GL(V1⊕V2)via g↦→ ϱ1(g)⊕ϱ2(g). For ϱ∈Rep(G), let

ϱ∗:G→GL(V

G(ϱ)∗)be the representation defined by ϱ∗(g):= (ϱ(g)∗)−1, i.e., the

group homomorphism corresponding to the action

G×V

G(ϱ)∗−→ V

G(ϱ)∗(1)

(g,ϕ)↦−→ ϕ◦ϱ(g)−1

A.2.2 Characters, Roots, Weights

Assume henceforth that Gis reductive, we have chosen a Borel B⊆Gand a maximal

torus Gr

m∼

=T⊆B.

A multiplicative (resp. additive) character of an algebraic group His a homomor-

phism of algebraic groups H→Gm(resp. H→Ga). Conversely, a multiplicative

(resp. additive) one-parameter-subgroup (also called 1-psg for short) is a homomor-

phism of algebraic groups Gm→H(resp. Ga→H). When we say character or

1-psg, we refer to the multiplicative version. The set of characters X(H)and the set

132

of 1-psgs ˇ

X(H)become abelian groups under pointwise multiplication. There are

canonical isomorphisms

Zr−→ X(Gr

m)Zr−→ ˇ

X(Gr

m)(2)

λ↦−→ χλγ↦−→ ˇ

χγ

where χλand ˇ

χγare defined as

χλ:Gr

m−→ Gmˇ

χγ:Gm−→ Gr

t↦−→ tλ:=tλ1

1···tλr

rx↦−→ tγ:= (tγ1, . . . , tγr).

For (γ,λ)∈ˇ

X(H)×X(H)we define ⟨γ,λ⟩∈Zto be the integer corresponding to

λ◦γ∈X(Gm)∼

=Z. In other words, g⟨γ,λ⟩= (gγ)λfor all t∈C×. It is well-known

[TY05, 22.5.2] that X(T)∼

=Irr(T)in the sense that any ϱ∈Rep(T)decomposes as a

direct sum of weight spaces

V(ϱ)α={v∈V(ϱ)|∀t∈T:t.v=tαv}.

If ϱ∈Rep(G), we can also regard V(ϱ)as a T-module and denote by V(ϱ)αthe

corresponding weight space. We call Λ(G):=X(T)the weight lattice of G.

To avoid an introduction to the theory of Lie algebras ([Hum98, Chapter III] or

[TY05, 23]), we deviate from [Hum98] in defining α∈Λ(G)to be a root of Gwith

respect to Tto be a if there exists an additive 1-psg εα:Ga→Gsuch that

∀t∈T:∀x∈C:t·εα(x)·t−1=εα(tαx).

This definition of root is equivalent to the one given in [Hum98, 16.4]: Indeed, one

direction of this equivalence is stated in [Hum98, 26.3] and the proof actually verifies

the other direction as well.

Denote by Φ:=Φ(G,T)⊆Λ(G)the set of roots of Gwith respect to T. See

[Hum98, 27] for a proof that Φis a so-called abstract root system and [Hum80, III.9.2] or

[TY05, 18] for general facts about abstract root systems. For α∈Φ, set Uα:=εα(Ga).

We call Φ+:={α∈Φ|Uα⊆B}the positive roots of Gwith respect to B.

A.2.3 Example. We continue Example A.2.1 and keep the notations already estab-

lished. We identify Λ(G) = X(T)with Znin the above way. For i,j∈[n]and i=j,

denote by αij ∈Znthe vector with the value 1 in position i, the value −1 in position j

and 0 elsewhere. We claim that these weights are the roots of Gwith respect to T.

For i,j∈[n], we define Eij ∈Cn×nto be the matrix with:

xrs(Eij) = {1 ; (r,s) = (i,j)

0 ; (r,s)= (i,j)

133

With these notations, we define the map

εij :Ga=C−→ G

x↦−→ In+x·Eij

It is elementary to check that this is a morphism of algebraic groups and furthermore,

for any x∈Cand any diagonal matrix t= (t1, . . . , tn)∈T,

t·εij(x)·t−1=t·(In+x·Eij)·t−1=tInt−1+x·tEijt−1

=In+x·ti

tj·Eij =εij(tαij x).

We therefore know that the αij are certainly roots. With more prelude and the the-

ory of Lie algebras, one can see rather directly that these are, in fact, all the roots.

See [Kra85, Bemerkung 4, III.1.4] or [GW09, Theorem 2.4.1] for a quite elementary

treatment.

The group Uij :=Uαij is now the image of εij and from the definition we can see

that Uij ={In+t·Eij ⏐⏐t∈C}consists of matrices with 1 on the main diagonal and

zeros everywhere else except at position (i,j). Since we chose Bto be the group of

upper triangular matrices, we can see that αij ∈Φ+if and only if i<j.

It can be shown that there exists a (unique) subset ∆⊆Φof linearly independent

weights with the property that for all α∈Φ+, there are nonnegative integers cδsuch

that α=∑δ∈∆cδδ. See [Hum80, III.10.1] or [TY05, 18.7.4] for a proof of this result. A

root belonging to ∆is called a simple root and|∆|is called the rank of G.

For every root α∈Λ(G) = X(T), there is a unique dual root ˇ

α∈ˇ

X(T)satisfying

the two conditions

•⟨ˇ

α,α⟩=2 and

•∀β∈Φ:β−⟨ˇ

α,β⟩·α∈Φ.

This is part of the root system axioms, see [Hum80, III.9.2] or [TY05, 18.2.1]. Note

that the map β↦→ β−⟨ˇ

α,β⟩·αis referred to as σαin [Hum80], as sα,ˇ

αin [TY05] and

as ταIn [Hum98]. It is called the reflection relative to α. It maps αto −αand leaves a

certain hyperplane Hαinvariant. We then call

Λ+(G):={λ∈Λ(G)⏐⏐∀α∈Φ+:⟨ˇ

α,λ⟩≥0}

the dominant weights of G, the notation being justified in that they lie on the side of

Hαwhich we have marked as “positive” by our choice of Φ+, which in turn is based

on the choice of B. See [TY05, 18.11.7] for a proof that Λ+(G)is a finitely generated

semigroup.

134

A.2.4 Example. We continue Example A.2.3. For the set ∆, we choose in this case the

roots αi:=αi,(i+1)for i∈[n−1]. There are n−1 of these roots, they are linearly

independent and for i<j, we have

αij =αi+αi+1+···+αj−1.

The rank of GLnis therefore n−1.

Given any root α∈Zn∼

=X(T), we define its dual to be the 1-psg corresponding to

the same tuple via (2). With this definition,⟨ˇ

α,α⟩corresponds simply to the ordinary

scalar product Zn×Zn→Zgiven by (α,β)↦→ αtβ. We identify αand ˇ

α. Clearly,

⟨αij,αij⟩=αijtαij =2.

Let λ∈Zn. For i<j, we have λi≥λjif and only if⟨αij,λ⟩=λi−λj≥0. Hence,

the dominant weights of GLnare precisely the weakly descending vectors of integer

numbers – such a vector is called a generalized partition.

We remark that in this case, it is easy to see that the semigroup Λ+(GLn)is finitely

generated: The vectors

ω1:= (1, 0,0, . . . , 0),

ω2:= (1, 1,0, . . . , 0),

ωn:= (1, 1,1, . . . , 1)

(3)

form a generating set together with −ωn.

We denote by Λr(G)the N-span of Φ+. The group Λr(G)generated by Λr(G)is

equal to the Z-span of all roots and is sometimes called the root lattice of G.

For two characters λ,µ∈X(T)we write λDµif λ−µ∈Λr(G).

A.2.5 Example. Still continuing Example A.2.4, we remark finally that the root lattice

of G=GLnconsists of all λ∈Znwith the property that the sum of all entries

vanishes, i.e.,|λ|:=λ1+···+λn=0. Furthermore,

Λr(G) ={λ∈Zn|∀k∈[n]:λ1+···+λk≥0,|λ|=0}.

Indeed, if λ=∑n−1

k=1ckαkis a sum of simple roots with ck∈Z, then λk=ck−ck−1.

Hence, ck∈Nfor all k∈Nif and only if

λ1+···+λk=c1+ (c2−c1) + ···+ (ck−ck−1) = ck≥0.

For dominant weights λ,µ∈Λ+(G), the dominance order is in agreement with

the following combinatorial definition of dominance order for integer partitions: Two

integer partitions λ= (λ1, . . . , λn)and µ= (µ1, . . . , µn)satisfy λDµif and only if

|λ|=|µ|and for all k≤n, we have λ1+···+λk≥µ1+···+µk.

135

Finally, we sum up the characterization of the irreducible representations of Gin

the following theorem:

A.2.6 Theorem. Let ϱ∈Irr(G)and V:=V(ϱ). There exists a unique dominant

weight λ∈Λ+(G)such that:

(1) As a T-module, V=⨁µEλVµ.

(2) dim(Vλ) = 1. Elements of Vλare called highest weight vectors.

(3) Uacts trivially on Vµif and only if µ=λ.

(4) For α∈Φand µ∈Λ(G), we have Uα.Vµ⊆⨁d∈NVµ+dα.

This defines a bijection Irr(G)∼

=Λ+(G). We henceforth identify the two.

Proof. The main statement as well as (1) to (3) can be found in [Hum98, Theorem 31.3]

together with the statement [Hum98, 31.4] which proves that each dominant weight

has a corresponding irreducible representation. Part (4) is [Hum98, 27.2].

For ϱ∈Rep(G), the module V:=V(ϱ)decomposes as

V=⨁

λ∈Λ

+(G)

V(λ)

where V(λ)∼

=V(λ)⊕nλfor certain nλ∈N. We call V(λ)the isotypical component of

weight λof V. When f∈V(λ), we call wt(f):=λthe weight of f.

A.2.3 The Coordinate Ring of an Algebraic Group

We recite a famous, general result about the weights that appear in the coordinate

ring of a reductive, affine, algebraic group:

A.2.7 Theorem (Algebraic Peter-Weyl Theorem). Let Gbe a reductive, affine, alge-

braic group. The group G×Gacts on Gfrom the left via (g,h).x=gxh−1. Consider

ψ:⨁

λ∈Irr(G)

V(λ)∗⊗V(λ)−→ C[G]

f⊗v↦−→ f◦ωv

where ωv:G→V(λ)is the morphism g↦→ g.v.

Then, ψis an isomorphism of G×G– modules, where the action of the left and

right factor of G×Gis on the left and right tensor factor, respectively.

Proof. See [TY05, 27.3.9]. We give a quick remark about the action. Recall from (1)

that g∈Gacts on f∈V(λ)∗by g.f=f◦g−1and recall from Remark A.1.5 that the

induced action on the coordinate ring of Gis such that for ϕ∈C[G]and g,h,x∈G

136

we have ((g,h).ϕ)(x) = ϕ(g−1xh). Let λ∈Irr(G),f∈V(λ)∗,v∈V(λ)and g,h∈G.

Then, the following holds for all x∈G:

((g,h).ψ(f⊗v))(x) = ((g,h).(f◦ωv))(x) = f(ωv(g−1xh)) = f(g−1xh.v)

= (f◦g−1)(x.(h.v)) = (( f◦g−1)◦αh.v)(x) = ψ((g.f)⊗(h.v))(x).

Hence, (g,h).ψ(f⊗v) = ψ((g.f)⊗(h.v)) as claimed.

A.2.8. We summarize Examples A.2.1 and A.2.3 to A.2.5 as follows: The dominant

weights for the reductive group GLnare the generalized partitions

Λ+

n:={λ∈Zn|λ1≥ ··· ≥ λn}.

We denote by V(λ)the irreducible GLn-module corresponding to the highest weight

λ∈Λ+

n. Every GLn-module Vdecomposes as V=⨁λ∈Λ

nV(λ)nλfor certain nλ∈

A.2.9 Proposition. The coordinate ring of GLn⊆Cn×nhas a natural Z-grading be-

cause it is the localization of a polynomial ring at the homogeneous polynomial detn.

For d∈Z, the isomorphism from Theorem A.2.7 restricts to

C[GLn]d∼

=⨁λ∈Λ

|λ|=d

V(λ)∗⊗V(λ).

Proof. Let λ∈Λ+

n,d:=|λ|and f⊗v∈V(λ)∗⊗V(λ). By Theorem A.2.7 we only

have to show that f◦ωvis homogeneous of degree d. Given t∈C×, Theorem A.2.6

implies that δt:=diag(t, . . . , t)acts on a vector v∈V(λ)as δt.v=td·vbecause for

all µEλ, we have|µ|=|λ|=d(Example A.2.5). Hence,

f(ωv(t·g)) = f(ωv(gδt)) = f(g.δt.v) = f(g.(td·v)) = f(td·(g.v))

=td·f(g.v) = td·f(ωv(g)).

Note that the fourth and fifth equality are due to the fact that v↦→ g.vand fare both

linear maps.

The module V(λ)∗is also an irreducible module. More precisely, for any dominant

weight λ= (λ1, . . . , λn)∈Λ+

nwe define λ∗:= (−λn, . . . , −λ1)∈Λ+

n. Then,

A.2.10 Proposition. Let λ∈Λ+

n. Then, V(λ)∗=V(λ∗)is an irreducible module.

Proof. This follows from Theorem A.2.6 and (1).

137

A.2.11. Proposition A.2.10 is a special case of a general principle. We denote by Bthe

set of all Borel subgroups of G. By [Hum98, Proposition A in 24.1], the Normalizer

N:=NG(T)of Tpermutes the elements of Btransitively. For G=GLnand T

the subgroup of diagonal invertible matrices, note that Nis the semidirect product

SnnT, where the permutation group Snis embedded in GLnas the subgroup of

permutation matrices.

The induced action of the Weyl group S:=N/Tis free, so Sparametrizes exactly

the possible choices of a base for the root system with respect to the torus T. The

action of Son Bis given by B↦→ Bq:=q−1Bq for q∈N.

Let τ∈Nbe such that B∩Bτ=T. Note that in the case of G=GLn, this is simply

the permutation that maps (1, . . . , n)to the reverse vector (n, . . . , 1), because Bis the

set of upper triangular matrices and B∩Bτ=Tif and only if Bτis the set of lower

triangular matrices.

The image of τin Sis called the longest Weyl element. In the literature, it is often

denoted by w0. With the terminology from [Spr08], w−1

0has the same length as w0and

by [Spr08, Corollary 8.3.11], an element with this property is unique, hence w0=w−1

It follows that τ=τ−1.

The Weyl group acts on the weights X(T)as follows: For any χ∈X(T)and q∈N,

we define the character q.λvia the rule (q.λ)(t):=λ(qtq−1). Then, the longest Weyl

element maps highest weights to highest weights [Hum98, 31.6]. More precisely, we

have V(λ)∗=V(−τ.λ). In the case of G=GLn, we have −τ.λ=λ∗.

In particular:

A.2.12 Proposition. Let Vbe a G-module. Then, VG∼

=(V∗)G.

Proof. By Theorem A.2.6,V=⨁λ∈Λ

+(G)V(λ), so VG=V(0). By the above Para-

graph A.2.11,V∗=⨁λ∈Λ

+(G)V(−τ.λ)where τdenotes the longest Weyl element.

Since −τ.λ=0 is equivalent to λ=0, we have (V∗)G∼

=VG.

We end by proving a Lemma that we will require in the next section and which is

of independent interest:

A.2.13 Lemma. Let Vbe a G-module (or more generally, a factorial G-variety) and

f∈C[V]. Assume that f=fk1

1··· fkr

ris the decomposition of finto irreducible

factors. Then, fis a highest weight vector of weight λif and only if the fiare highest

weight vectors of weight λiwith λ=k1λ1+···+krλr.

Proof. If the fiare weight vectors, then fis still U-invariant and it is easy to see that

the weight of fmust be equal to k1λ1+···+krλr.

For the converse, assume that fis a highest weight vector of weight λand denote

by Z:=Z(f)⊆Vthe vanishing set of f. Note that Zis B-invariant, so we have

138

an action α:B×Z→Zinduced by the action of Bon V. Let Zi:=Z(fi), then

Z=Z1∪. . . ∪Zris the decomposition of Zinto its irreducible components. Let

αi:B×Zi→Zbe the restriction of αto Zi. Since Bis connected, B×Ziis irreducible

and Ziis contained in the image of αi– therefore, B.Zi=αi(B×Zi) = Zi. Hence,

the Ziare invariant under B. It follows that B.fi⊆C·fiso Bacts by a character on fi.

This means that fiis a highest weight vector of some weight λi. Consequently, we

must have λ=k1λ1+···+krλr.

A.3 Polynomial Representations

Let W∼

=Cnbe a complex vector space with a chosen basis. The action of GLnon a

module V∼

=CNis a morphism ϱ: GLn→GLNof algebraic groups. Hence, it can

be described by N2regular functions on GLn. In many cases we are interested in,

these functions are polynomials in the entries of an n×nmatrix, i.e., ϱextends to a

morphism Endn→EndNof affine spaces. In other words, the action of GLnextends

to an action V×Endn→V. We call such a representation a polynomial one.

A.3.1 Definition. There is a partial ordering ⊑on Λ+

n, called the inclusion, defined

as follows: We have µ⊑λif and only if µi≤λifor all 1 ≤i≤n.

Note that µ⊑λimplies µEλ, but not the converse. We have

Λ+

n:={λ∈Λ+

n⏐⏐λ⊒0}={λ∈Zn|λ1≥. . . ≥λn≥0}⊆Nn,

the set of partitions of the number n. We will show:

A.3.2 Proposition. Let λ∈Λ+

n, then V(λ)is polynomial if and only if λ∈Λ+

When Vis any GL(W)-module, we can decompose it as V∼

=⨁λ∈ΛV(λ)⊕nλfor

certain nλ∈N. We then write

V⊒0:=⨁

λ∈Λ

λ⊒0

V(λ)⊕nλ⊆V.

We call V⊒0the polynomial part of V. It then follows:

A.3.3 Corollary. A GLn-module Vis polynomial if and only if V=V⊒0.

We require an auxiliary lemma for the proof of Proposition A.3.2.

A.3.4 Lemma. The inclusion GLn⊆Endnis an open, GLn-equivariant immersion of

varieties under the operation of GLnacting by multiplication from the left on both

affine varieties. It induces an inclusion of their respective coordinate rings which

satisfies C[Endn] = C[GLn]⊒0=⨁λ∈Λ

nV(λ)⊗V(λ)∗.

139

Proof. By [Lan12, Formula (6.5.1)], we get the last equality in

C[End(W)]d∼

=C[W⊗W∗]d∼

=Symd(W⊗W∗)∼

=⨁

0⊑λ∈Λ

λ1+···+λn=d

V(λ)⊗V(λ)∗.

By summing over dand applying Theorem A.2.7, we obtain

C[End(W)] ∼

=⨁

λ∈Λ

V(λ)⊗V(λ)∗=C[GL(W)]⊒0

by the definition of the polynomial part.

Proof of Proposition A.3.2.By Theorem A.2.7, the module C[GLn]contains a unique

highest weight vector of weight µfor every µ∈Λ+

n, up to a scalar. Let fbe the

highest weight vector of weight λ= (λ1, . . . , λn). By Lemma A.3.4, we have λ∈Λ+

if and only if fextends to a regular function on Endn, i.e., it is a polynomial. Let

f=q·detk

nwith k∈Zand q∈C[Endn]not divisible by detn. We have to show that

k≥0 is equivalent to λn≥0. In fact, we show that k=λn.

Denote by δ:= (1, . . . , 1)∈Λ+

n, then wt(detn) = δ. We claim that µ:=wt(q)

satisfies µn=0. Indeed, otherwise we’dhave µ−δ∈Λ+

nand Lemma A.2.13 would

imply that qis divisible by the (unique) weight vector of weight δ, which is detn.

Consequently, λn=µn+k=k.

140

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List of Symbols

⊑Inclusion order on generalized partitions (page 137).

≡Equivalence of complexity measures (page 36).

∇(P)Row vector of partial derivatives of P(page 59).

[m]The set of natural numbers from 1 to m(page 16).

X//GQuotient of a G-variety Xby G(page 125).

P∼QThe form Pis in the GL-orbit of the form Q. (page 108).

αXaction morphism of a variety with semigroup action (page 123).

A♯Adjugate of A, transpose of the cofactor matrix (page 80).

APAnnihilator scheme (page 82).

Ass

PSemistable part of ˆ

AP(page 82).

AtTranspose of the matrix A(page 41).

APAnnihilator of a form P(page 61).

bdc(P)Binary determinantal complexity of a polynomial P(page 21).

bndThe generic binomial (page 105).

cc(P)Circuit complexity of a polynomial Pwith coefficients in a fixed ring

(usually a field). (page 10).

ccR(P)Circuit complexity of a polynomial Pwith coefficients in R(page 10).

Cdd-th Chow variety, orbit closure of mnd=x1x. . . xd(page 49).

coneQ(S) Rational cone generated by a subset S⊆Zn(page 47).

CJtKThe ring of formal power series with complex coefficients (page 77).

C[X]Coordinate ring of an affine variety X(page 123).

DdThe orbit closure of detd(page 45).

DϱDifferential of a morphism ϱbetween varieties (page 79).

149

∂iPThe i-th partial derivative of a polynomial P(page 58).

∂wPPartial derivative of a form Pin direction w(page 58).

Pd,mThe orbit closure of ppd,m(page 45).

dc(P)Determinantal complexity of a polynomial P(page 18).

dc(P)Determinantal border complexity of a polynomial P(page 37).

detdThe determinant polynomial of a generic d×dmatrix (page 16).

doc(P)Determinantal orbit complexity of a polynomial P(page 36).

doc(P)Determinantal orbit border complexity of a polynomial P(page 37).

dom(ϖ)Domain of the rational map ϖ(page 81).

End(W)The linear endomorphisms of a vector space W(page 35).

End(W,L)Endomorphisms on Wthat map into the subspace L⊆W(page 89).

G◦Identity component of an algebraic group G(page 125).

Γ(φ)Graph of a rational map φ(page 81).

GPStabilizer of a homogeneous polynomial P(page 41).

GaThe additive group (page 128).

GL(W)The general linear group of invertible endomorphisms of W(page 124).

GLnThe group of invertible, complex n×nmatrices (page 124).

glnLie algebra of GLn(page 79).

gl(V)Lie algebra of GL(V)(page 79).

GmThe multiplicaive group (page 128).

HCmHamilton Cycle polynomial (page 31).

InThe n×nidentity matrix (page 47).

idWThe identity map on W(page 59).

im(a)Image of an endomorphism a(page 63).

Irr(G)All equivalence classes of irreducible G-representations (page 130).

|λ|Sum of all entries of an integer vector λ∈Zn(page 46).

ℓ(λ)Length of a partition λ(page 46).

LUsually a linear subspace of W, often a linear subspace of some hyper-

surface Z(P)⊆W(page 61).

λ∗Negative reverse of a parttion λ(page 135).

Λ+(G)Dominant weights of an affine algebraic group G(page 132).

150

Λ+(Z)Dominant weights in the coordinate ring of a GLn-variety Z(page 46).

Λ(G)Weight lattice of an affine algebraic group G(page 131).

Lie(G)Lie algebra of an algebraic group G(page 79).

Λ+

nGeneralized partitions, the dominant weights of GLn(page 135).

Λ+

nDominant weights of GLnthat yield polynomial representations, parti-

tions of n(page 137).

Λ(Z)Group generated by Λ+(Z)(page 47).

mndThe universal monomial mnd=x1···xd(page 49).

NPNullcone of the action of the stabilizer group on the space of Endomor-

phisms (page 62).

N(Z)Normalization of a variety Z(page 48).

ΩP,Ω(P)Orbit of a homogeneous polynomial (page 39).

ocP(λ)Orbit coefficient of Pat λ(page 42).

ΩP,Ω(P)Orbit closure of a homogeneous polynomial (page 39).

ocP(λ)Orbit closure coefficient of Pat λ(page 40).

∂ΩPBoundary of the orbit of a homogeneous form (page 60).

PUsually a set of polynomials (page 35).

ΦUsually a root system (page 131).

permThe permanent polynomial of a generic m×mmatrix (page 16).

P≤Q P is a projection of Qover a fixed ring (page 14).

P≃QThe two polynomials Pand Qare equivalent over a fixed ring (page 35).

P≤RQ P is a projection of Qover the ring R(page 14).

P≃RQThe two polynomials Pand Qare equivalent over the ring R(page 35).

poly The set of polynomially bounded functions N→N. (page 12).

poly(n)Expressions that are polynomially bounded in n∈N. (page 12).

ppd,mPadded permanent polynomial (page 39).

PσThe polynomial that arises from Pby the substitution σ(page 14).

QUsually a set of polynomials (page 35).

Rep(G)All equivalence classes of G-representations (page 130).

R(G)Radical of an algebraic group G(page 128).

rk(a)Rank of an endomorphism a(page 59).

151

Ru(G)Unipotent radical of an algebraic group G(page 128).

R[x]The polynomial ring in a countably infinite set xof variables (page 10).

⟨S⟩Free abelian group generated by S(page 46).

SmThe symmetric group on msymbols (page 16).

Sat(S)Saturation of a semigroup S(page 46).

SLnGroup of complex n×nmatrices with determinant 1, the special linear

group (page 129).

slnLie algebra of SLn(page 81).

supp(P)Support of a polynomial P(page 13).

SymkWThe k-th symmetric power of the vector space W(page 49).

tr(A)The trace of a square matrix A(page 80).

VA complex vector space, usually V=C[W]d(page 57).

V⊒0Polynomial part of a GLn-module V(page 137).

V(ϱ)Vector space of a representation ϱ(page 129).

G(ϱ)Vector space of a G-representation ϱ(page 129).

ωPOrbit map of a point/polynomial P(page 61).

ϖPRational projective orbit map of a point/polynomial P(page 61).

WA complex vector space of dimension n(page 57).

xA countably infinite set of formal variables (page 10).

X(G)Character group of G(page 130).

X(G)One-paramter-subgrups of G(page 131).

Z(P)Affine vanishing set of a homogeneous form P(page 63).

152

Index

#P (complexity class):15

1-psg: 132

additive group: 130

adjugate: 81,91

affine cone: 60

algebraic group: 126

Lie algebra: 80

algebraic monoid: 126

algebraic semigroup: 125

morphism: 125

Alon-Tarsi conjecture: 46

annihilator: 61,91,118

approximation order: 79

approximation path: 79

linear: 79

order: 79

arithmetic circuit: 10

arithmetic complexity measure: 35

binary determinantal complexity: 21

binary variable matrix: 21

binomial: 107

blowup: 83,85

smooth: 85

border complexity: 37

Borel subgroup: 130

boundary: see also orbit

categorical quotient: 127

center: 83

character: 132

characterized by its stabilizer: 43

Chow variety: 47,49,67

circuit: 10

complexity: 10

computes: 10

constant-free: 30

size: 10

skew: 30

weakly skew: 17

closed: 65

closure: see also orbit

complete: 16

complexity: see also circuit

complexity measure: 35

compression space: 91

computation gate: see also gate

computes: see also circuit

concise: 58,90,107

constant-free: see also circuit

degeneration: 58

linear: 79

degree: 131

depth: 10

determinant: 16

determinantal border complexity: 37

determinantal complexity: 18

determinantal orbit border

complexity: 37

153

determinantal orbit complexity: 36

digraph: 22

value: 22

weight: 22

dimension datum: 43

domain: 82

dominant weights: 134

endomorphism

image: 63

rank: 59

equivalent: see also polynomial

G-variety: 125

morphism: 125

gap: see also semigroup

gate

computation gate: 10

input gate: 10

general linear group: 38,126

generalized partition: 135,137

inclusion: 139

geometric quotient: 128

good quotient: 127

graph: 83

highest weight vectors: 136

homomorphism: 132

image: see also endomorphism

inclusion: see also generalized partition

input gate: see also gate

invariant: 125

irreducible: see also representation

isotypical component: 136

Kronecker product: 81

latin square: 46

length: see also partition

Lie algebra: see also algebraic group

linear: see also approximation path, see

also degeneration

linear subspace

maximal: 62

semistable: 63

unstable: 63

linearized: 129

linearly equivalent: see also polynomial

maximal: see also linear subspace

module: 131

morphism: see also algebraic semigroup

multiplicative group: 130

nonuniform: 13

NP (complexity class):15

nullcone: 62,128

occurrence obstructions: 45

one-parameter-subgroup: 132

orbit: 39,125

boundary: 61

closure: 39

orbit closure coefficient: 40

orbit coefficients: 42

order: see also approximation path

order of approximation: 79

P(complexity class):12

p-family: 12

p-projection: 14

padded permanent: 39

partial derivative: 58,80

partition: 139

length: 46

path value: 22

path weight: 22

permanent: 16

plethysm: 69

polynomial: 139

equivalent: 35

154

linearly equivalent: 110

polynomial part: 67,139

polynomially bounded: 12

polynomially equivalent: 36

polynomially many: 12

polystable: 60

positive: see also root

power series: 77

projection: 14

radical: 130

rank: see also endomorphism, see also

reductive group

reductive: 57,130

reductive group

rank: 134

reflection: 134

representation: 131

irreducible: 132

root: 133

positive: 133

simple: 134

root lattice: 135

saturated: 46

saturation: 46

semigroup

gap: 46

semisimple: 130

semistable: 62,see also linear subspace,

84,129

simple: see also root

singular subspace: 98

size: see also circuit

skew: see also circuit

smooth: see also blowup

stabilizer: 125

stable: 125

subrepresentation: 132

support: 13

support matrix: 26

symmetric power: 49

toric variety: 49

torus: 130

trace: 81

transpose: 41

uniform: 13

unipotent radical: 130

unit group: 126

universal: 18

universal monomial: 49,67

unstable: 62,see also linear subspace

value: see also digraph

VNP (complexity class):14

VP (complexity class):12

weakly skew: see also circuit

weight: see also digraph, 136

weight lattice: 133

weight spaces: 133

Weyl group: 138

155