F om symme ic unc ions o
qubi s
F om symme ic unc ions o
qubi s
Ou pu pose:
Explain how o compu e he Hilbe se ies o he algeb as o co a ian s o (pu e)
qubi s sys ems.
Why hese algeb as a e ele an ?
The (pu e) qubi sys ems a e ega ded as mul ilinea o ms on which ac s a
p oduc o linea g oups (SLOCC: S ochas ic Local Ope a ions and he Classical
Communica ion).
The knowledge o he co a ian s allows:
- (In p inciple) o desc ibe he s uc u e o he o bi s.
- Cons uc en anglemen mono ones and measu e o en anglemen .
F om symme ic unc ions o
qubi s
Ou plan:
Lec u e I: In oduc ion o symme ic unc ions (Jean-Gab iel Luque)
We will p esen he main ool: he symme ic unc ions.
Lec u e II: Symme ic unc ions and cha ac e s o he symme ic g oups.(JGL)
We will explain why he symme ic unc ions encodes he cha ac e s o he symme ic
g oups.
Lec u e III: Ve ex ope a o and K onecke coe icien s (Jean-Y es Thibon)
We will explain he links be ween he K onecke coe icien s and he symme ic unc ions.
Lec u e IV: Hilbe se ies o he algeb as o in a ian s o a k-qubi s (pu e) sys em. (JYT)
We will explain how o use hese ools o compu e he Hilbe se ies.
F om symme ic unc ions o
qubi s
Lec u e I
In oduc ion o symme ic unc ions
Jean-Gab iel Luque
Symme ic unc ions
De ini ion:
Polynomials in se e al a iables (alphabe X={x1,...,xn,...}) which a e in a ian unde
pe mu a ions o he a iables.
Sym(X) : algeb a o symme ic polynomials o he alphabe X
Example
The monomial unc ions:

Symme ic unc ions
De ini ion:
Polynomials in se e al a iables (alphabe X={x1,...,xn,...} which a e in a ian unde
pe mu a ions o he a iables.
Sym(X) : algeb a o symme ic polynomials o he alphabe X
Example
The comple e unc ion hn is he sum o all he monomials o deg ee n
Symme ic unc ions
De ini ion:
Polynomials in se e al a iables (alphabe X={x1,...,xn,...} which a e in a ian unde
pe mu a ions o he a iables.
Sym(X) : algeb a o symme ic polynomials o he alphabe X
Example
The elemen a y unc ions
Symme ic unc ions
De ini ion:
Polynomials in se e al a iables (alphabe X={x1,...,xn,...} which a e in a ian s unde
pe mu a ion i he a iables.
Sym(X) : algeb a o symme ic polynomials o he alphabe X
Example
The powe sums
Bases
Suppose ha X={x1,...,xn,...} is an in ini e alphabe .
Mul iplica i e bases:
Non mul iplica i e bases:
Comple e unc ions
Powe sums
Elemen a y unc ions
Monomial unc ions Schu unc ions
Di e ences o alphabe s

P oduc o alphabe s
A powe sum o a p oduc o alphabe s is he p oduc o he powe sums.
Hence,
Scala p oduc and ep oducing
Ke nel (1)
Conside a scala p oduc :
Wi h a pai o bases in duali y
The ep oducing ke nel associa ed o { , } is a mul i a ia e se ies on wo alphabe s de ined by
Why his se ies is called a ep oducing ke nel?
Scala p oduc and ep oducing
Ke nel (2)
As a consequence, i one has
Fo any pai o bases in duali y:
Usual scala p oduc and inne
p oduc (1)
We se
Since,
This is he ep oducing ke nel o he usual scala p oduc de ined by
O equi alen ly,
The usual inne p oduc is de ined by
Usual scala p oduc and inne
p oduc (2)
W i e
Since
The coe icien o in is
Mo e gene aly

O hogonalisa ion o comple e
unc ions
One apply he G amm-Schmid o hogonalisa ion p ocess o he comple e unc ions o
he in e se o de o dominance
Example o he deg ee 4:
.....
By de ini ion he a e o hogonal. This basis is called he Schu basis.
An o hono mal basis (1):
Semi-s anda d ableaux
Semi-s anda d (Young) ableaux o shape
We ill up he nodes o he shape wi h non-nega i e in ege s such ha he en ies a e
s ic ly inc easing along each column and jus non-deg easing along each ow.
no no ok
An o hono mal basis (2):
Schens ed algo i hm
Schens ed algo i hm: cons uc a semi-s anda d ableau om a sequence o in ege s
I e a ion o he inse ion:
Inse ion o an in ege n in o a ableau
1) Fi s y o inse on he i s line.
......
......
...........m n
I m< n he esul is ob ain by glueing n a he end o he line
2) o he wise, le p he smalles in ege o he line s ic ly g ea e ha n
...........
...........
.......p........m n
Replace he i s occu ence o p by n and y o add p in o he nex line
...........
........... p
.......n........m
An o hono mal basis (3):
Schens ed algo i hm, example
Conside he sequence 42214163311
4
4
42 2
4
22
4
4 42 2
221 12 12
4
2
124
4 4
2 2 2
1241 1 1 4
4
22
1146
4 4
22 224
11463 1136
4 4
224 2246
11363 1133
4 4 44
2246 22463 2236
11331 1113 1113
44 44 446
2236 22363 2233
11131 1111 1111
Schu o m
Hence
whe e
a e he Kos ka numbe s
Example:

h o Schu
Since a e in duali y
The Schu basis being o hono mal, one has
implies
Conclusion
+ We ha e shown how o use gene a ing unc ion o
+ Compu ing changes o bases.
+ De ining he scala s anda d scala p oduc and he s anda d inne p oduc .
+ We ha e de ined Schu unc ion by o hogonalizing he comple e unc ions.
+ We ha e p o ed ha he Schu unc ions a e o hono mal o he s anda d scala
p oduc .