Pub
.
Ma
.
UAB
Vol
.
30
N3
1
Maig
1986
ON
THE
POINCARE
SERIES
OF
H*(GI,(2,2n),Z/2)
G .R
.
Chapman
INTRODUCTION
.
Le
G
be a
ini e
g oup, and
A
a
Noe he ian
G- ing
.
Then
E ens
[3],
Venko
[11]
show ha
H*(G,A), Che
cohomology
ing o
G
wi h
coe icien s
in
A,
is
ini ely
gene a ed
.
The
p oo s
a e
essen ially
non-cons uc i e,
and
gi e
li le
in o ma ion
conce ning
Che
deg ees
in
which
Che
ing
gene a o soccu
;
a
ques ion
i s
aised
by
Johnson
[6]
.
Explici
desc ip ions
o
p oduc
s uc u es
o
cohomology ings
a e
no
easy o
ob ain,
a
majo
di icul ybeing
o
de e mine
when
a
se
o
gene a o s
is
comple e
.
In
his
pape ,
we
exhibi
a
ci cums ance
in
which
his
di icul y
may
be
o e come,
and
gi en
an
example
.
Le
p
be
a
p ime,
and
le
Gp,(H*(G,A)]p
deno e
a
Sylow
p-subg oup
o
G,
H*(G,A)
espec i ely
.
Swan
[10]
shows
ha
when
Gp is
abelian,
hen
[H
*
(G,A)]
p
consis s
o Che
sub ing
o
H*
(G
P
,A)
ixedunde
he
ac ion
induced
by
inne
au omo phisms
o
G
.

Cbnside
Che
case
when
p=2,
G2 is
elemen a y
abelian,
and
A
is
Z/2,
he
in ege s
mod
2
wi h
i ial
G-ac ion
.
Then
H*(G
2
,Z/2)
is
isomo phic
o
R=(Z/2)[x1,
. .
.,x ],
a
polynomial ing in
inde e mina es
whe e
is
he
ank o
G
2
[7]
.
Consequen ly
H
*
(G,Z/2)
may
be calcula edas
a
ing
o
in a ian s
R
H
,
whe e
His a
g oup whose
o de
is
odd,
and
hence
cop ime
o
he
cha ac e is ic
o he
base
ield
o
R
.
In
exposi o y
a icles,
Sloane
[8]
and
S anley
[9]
discuss
.classical
in a ian
heo y,
in
which
he
base
ield
is
he
complex
numbe s
.
A
canonical
o m
is
gi en
o he
ing
o
in a ian s,
om
which
a
comple e
se
o
gene a o s
and
ela ions
may
be
de i ed
.
In
sec ion
2
we
indica e
how,
wi h
mino
modi ica ion,
hese
esul s
apply
o
he
si ua ion
desc ibed
abo e
.
In
sec ion
3
we
apply
hese esul s
o
G=GL(2,2
n
),
and
ob ain
an
exp ession
o he
Poinca é
Se ies
o
H*(GL(2,2n),Z/2)
.
The
addi i e
*
s uc u e
o H
(GL(2,2n),Z/2)
has
been
desc ibed
by
Aguadé
[1],
bu
he
knowledge
o
he
Poinca éSe ies
leads, ía he
canonical
o m
o
he
ing o
ín a ian s,
o
a
comple e
se
o
gene a o s
and
ela ions
.
Fo
n=2,
he
esul s
a e
well
known
[12]
.
Fo
highe alues
o n,
he
calcula ion
becomes
mo e complica ed,
and
he
esul s
o a
machine
compu a ionsa e
p esen ed
o n=3
.
I
would
like
o
hank
P .J
.
Webb
o
a
se ies
o
enligh ening
co espondences,
and
in
pa icula
o
indica ed
how he
B aue
li
could
be used o
p o e
he
e sion
o
Molien's
heo em
gi en
in
heo em
1 .
2
.
18
MOLIEN'S
THEOREM
AND
A
BASIS
OF
INVARIANTS
.
Le
F
be
a
ield,
H'a
ini e
g oup,
V
an
F(H)-module
wi h
F-basis
The
polynomial
ing
R =
F[x1,
. .
.,xn] is
{x1,
. .
.,xn}
and
cha ac e
X
.
g aded,
wi h k- h
componen
R
k
ha ing
F-basis
he
se
o
monomials
o
deg ee
k
in
XI,
.
.
.,x
n

(k>0)
.

Each
hc
H
induces
h
:V
+ V

by
h( )

= h
.
( cV), and o
each
j>0
a
map
h
j
:R
j }
R
j
de ined
by
This
makes
R
j
an
F(H)-module,
whose
cha ac e
we deno e
by X
j
.
Deno e
by R
H
he
sub ing
o
R
in a ian
unde
his
ac ion,
and le
a
j
=
dimF(RjH)
.
l
n

l
n
hj(xl
.
.
.x
n
)
=
h(x
j )
.
.
.h(x
n
)

(
l
+
.
.
.+
n = j)
"
In
he
classical
heo y,
whe e
Fis
aken
o
be
he
complex
numbe s
(C)
Molien's
Theo em
yíelds
an
explici
exp ession
o
£
aj j,
he
Poinca é
Se ies
o RH
.

Mo eo e ,
RH
is
a
Cohen-Macauley
ing
whichmeans
he
Poinca é
Se ies
may
be
w i en
k
i
i
II l
(1-
)
Consequen ly,
he e
exis
ee
in a ian s
1
,
. .
.,
.
(deg
i =
i )
and
ansien
in a ian s
9
11
. .
.,gk
(deg
g i =
u
i
)

such
ha
k
RH
=
1L
g,C[ 1,
. .
., ]
i=1
and
l,
.
.
., ,
k
a e
algeb aically
independen
o e
C
.
The
esul s
ske chedhe e
a e
discussed
mo e
ully
in
[8],
[9]
.
Now
suppose
ha
cha (F)~
H
.
The
ac
ha
R
H
is
Cohen-Macauley
ollows
di ec ly
om
[5]
P opn
13
p1033,
so
ha
as in
he
complex
case
k
R
H
]
=

LL
gi
F[
l , ".
.,
R
.
i=1
Molien's
heo em
may
be
modi ied
by
i s
assuming
ha
F
con ains
IHI- h
oo s
o
uni y
.
This
may
be achie ed
by
enso ing
up o a
sui able
ield
i
necessa y,
bu
does
no
e ec wha
ollows
.
Fo
h
¿H, le
B(h),
B(h
k
)
deno e
a
B aue
li
o h, hk
espec i ely,
and
B(X),
B(X
k
)
he
B aue
cha ac e s
o V, Rk
espec i ely
.
We
ha e
he
ollowing
e sion
o
Molien's
heo em
.
Theo em
1

I
cha
(F) i
HI,
hen
whe e
c(B(h))
is
he
cha ac e is ic
polynomial
o
B(h)
in
he
inde é mina e
.
PROOF

Le
n
l
"
"" .n
n
e

C
deno e he
eigen alues
o B(h)
.
Then
l
n
whe e
X
j
=

E

n
1 . .
.nn

.
1
+
. . .
+
n =
j
1
n
Bu
{n1
. .
.n
n
;
1
+
."
.+ n =
j}
is a se
o
eigen alues
o
B(hk),
so ha
X
j
=
B(X
k
)
(h)
.

(2)
Since
.
cha (F)j
H, he
o hogonali y
ela ions
o
B aue
cha ac e s
[2]
§18C
a e
simila
o
chose
o
o dina ycha ac e s
.
In
pa icula ,
i
ollows
ha
1
de (B(h))
j=Oaj j
.
7
T
heH

c(B(h))
[de (I
-
B(h) )]
-1
=
i, l(1-ni )-1
aj =
~
.

E
B(X
j
)(h)
.
heH
W
=
j E
0 X
j
j

(1)
Hence
by (1), (2)
and
(3),
and
he
heo em ollows
.
«0
0)
;
BE GF(2
n
)}
cyclic
g oup
o
o de
2
.
I
1
+C + N
iH i
1
.
He e
H
ac s
on
G2 by
E
hEH
3
.
THE
POINCARE
SERIES OF
H*(GL(2,2n),
Z/2)
.
Fo
n>1,
le
GF(2
n
)
deno e
he
ield
o 2
n
elemen s,
and
G be
CL(2,2
n
),
he
g oup
o
2x2
ma ices
wi h
en ies
in
GF(2
n
) .
A
Sylow
2-subg oup
G
2
o G
consís s
o he
ma ices
de (I-B(h) )
and
is
isomo phic
o
he
di ec
p oduc
o
n
copies
o
C2, he
0

ñ
1
hen
H is
cyclic
o
o de
2
n-1,
and
i N, C
deno e he
no malize
o
G
2
in G,
and
cen alize
-
o G
2
in G
espec i ely,
we
ha e
he
ex ension
(a

0

)

( 1

0)

a

1

0)

-

~1

a
2
0
)
0

o¡-
1

0

1

0

a

0

1
and
since
Au (G2)
=
CL(n,2), we ha e
a
monomo phism
O :H i
GL( i,2)
.
This
ís
discussed
mo e
ully
in
[4]
.
n
Le
P
n
( )
° j E0c
j+l
3
be
a
p imi i e,
i educible,
deg ee
n
polynomial
o e GF(2),
and,le
p be a
oo
o
P
n
( )
.
Then
H
is
ó 0

2n-1
gene a ed
by

(
0

ó-1)
whe e
6 =
11

.

Fu he ,
as a
ec o
space

o e
F2,
G
2
has
basis
Since
d
.
i
ollows
ha
0
(

o-

is
he
companion
ma ix
M o Pn
( ),
and
0
d
1
ha
0(H)
= M, he
g oup
gene a ed
by
M
.
Fo
1<i
<
n,
le
x
i
be
he
elemen
o H2
(G2,Z/2)
which
co esponds
unde
his
isomo phism
o
he
homomo phism
which
maps
o
1
i
j
= i-1
and
0
o he wíse
.
l
is
well
known
(see
e.g
.
[7])
ha
H*
(G2,Z/2)is, he
polynomial
ing
R =
GF(2)[x1,
. .
.,xnl
.
l
.
ollows
om
he
de ini ion
o
x,
ha
M
induces
a
ans o ma ion
so
ha ,
iden i ying
H
wi h
i s
image
unde
0,
we
H*
(G,Z/2)
as
he
ing o in a ian s
RH
.
whe e
B
deno es
he
B aue
li ,
and
c
he
cha ac e is ic
polynomial
.
To
simpli y
his
exp ession,
we
i s
no e
ha
i Q is
a
polynomial
o e
GF(2), hen
Q(y2)
_
[Q(y)J
2
.
Hence
i
Q( )
is
i educible
o deg ee
d,
he
oo s
o
Q( )
a e
o
he
o m
2
2d-1
{y,y
, . .
.,y

} .
Since
he
cha ac e is ic
polynomial
o
M
is
2
2
xi
+
xi+1 (l<i<n-1),
n
xn

i

j
El

cjxj
(0

°
-i
)(0

1

(ó
-1 °

°

(0
Tu ning
o
cohomology
we
no e ha
H
1
(G
2
,Z/2)
Z
Hom
(G
2
,Z/2)
.
By
heo em
1,
he
Poinca é
se ies
is
i+l
u
1~
(Ocicn-1),
(1
uj
0
1
may
calcula e
1
2n-2
de
B(Mi)

(4)
2n-1

1=0

c
B(M
i
)
P
n
( ),

í
ollows
ha

M

is
simila
o
diag

(u,V2,
. .
.,u
2n-1
),

and
n-1
ha
M
i
is
simila
o
diag
(pi,

,u í2

) .
To simpli y
(4),
conside
he
ac ion
o
Z/n
( he
in ege s mod
.
n)
on
Xn = {0,1,
. .
.,2n-2}
gi en
by
z(i)
=
esidue
o
2
zi
mod
.
2 n
-1
(zEZ/n,
¡EX
n
)
.
I
O b(i)
deno es
he
o bi
o
i
and
¡O b(i)l
= di
,
hen di
is
he
exponen
o
2 mod
.
e(i),
whe e
2
n
_1
(2
n
-1
,i)
is
he
exponen
o
P i
in
GF(2
n
)
.
Mo eo e ,
pi
is a oo o
an
i educible polynomial
o
deg ee d
i
o e GF(2)
.
Fo
each din, le
O
d
deno e a se
o
ep esen a i es
o
he o bi s o
size
d
.
I
n
is
a
p imi i e
complex
(2
n
-1) h
oo
o
uni y
co esponding
o
V
unde
he
B aue
li ,
we
ha e
j
de B(M
i
)
=
ni l
ni2
=
1
j
=0
n-1
j
d
i
-1
j
n/d
i
cB(M
i
)
=

n
(n
i2
- ) =

n
(n
i2
- )

(o<i<2n-2)
.
j
=o

J=O
Thus om
(4)
we ob ain
Theo em
2
.
The
Poinca é
Se ies
o
H*(GL(2,2n),Z/2)
is
1
E
(

d

)
2
n
-1
din
¡EOd
d-1

j
1[
(n2
i- )
n
/ d
j=o
whe e nand Od a e
de ined
abo e
.
We
exhibi
he Poinca é
Se ies
o
some
low
alues
o
n
.
An
explici
exp ession
seems ha d
o
ob ain
o
a bi a y
n
.
(i) l
n
=
2,
he
o bi s
o
X
2
a e
{0},
{1,2}
and
heo em
2
gi es
[ 1
+2
1
(n3
=1)
3
(1- )
2
(n- )(n2- )
as
he
Poinca é
Se ies
.
This
simpli ies
o
1- + 2
24
(1- )(1- 3)
as is
well
known
(see
e
.g
.
(12})
.
(íi)
I n = 3,
he
o bi s
o X3 a e {0},
{1,2,4},
{3,5,6}
so
we ob ain
1
1
+

3

+

3

}
(n7=1)
7
(1- )
3
(n- )(n2- )(n4- )
(n3- )(n5- )(n6- )
which
can
be
w i en
1-2 +
2
+
3
+
4
-2
5+ 6
(1_0
2
(1-
7)
(iii)
Fo
n
=
4,
he
o bi s
o X
4
a e
{0}
{5,10}, {1,2,4,8},
{3,6,9,12}
and
{7,11,13,14}
.
A
leng hy
calcula ion
shows
he
Poinca é
Se ies
is
1-2 +
2
-
3
+3
4
+
5- 6-
7
+ 8
-
.
9
-
10
+
ll
+3
12- 13
+
14
-2
15+ 16
(1- )2(1- 3
)(1- 15)
4
.

The
Cohomology
Rings
o
n=
2,3
.
As
obse ed
in
sec ion
2,
he
Poinca é
Se ies
(as
gi en
by
heo em
2)
may
be
w i en
in
he
o m
R
n
(1-
i)
í=1
hough
no
necessa ilyuniquely
.
Howe e ,
i
ee
in a ian s
can
be
ound
in
deg ees
1 ,

,
L
,
and
. ansien
in a ian s
in
deg ees
1
,
.
.
.,
k
hen
we
may
conclude
ha
hese
gene a e
he
en i e ing
.
(i)
When
n = 2, (6)
may
be
w i en
W i e
x,y
ins ead
o
xl,x2
.
Since
A=x
2
+xy+y
2
,
B=xy(x+y)
and

C=x
3
+
x2
y
+ y
3
a e
in a ian swi h
C
2
= A3 + B
2
+C.B,
i
ollows
ha
hese h ee
elemen sgene a e
H*(GL(2,22),Z/2)
as a
commu a i e
ing
o
exponen
2
subjec
o he
single
ela iongi en
.
(ii)
Fo
n
=
3,
he
Poinca é
Se ies
may
be
w i en
1
+
2 4
+3
5
+3 6
+
2
7
+ il
W i e x,y,z
o
x
l
,x
2 ,x
3
.
Sea ching
in
he
ing
o
in a ian s,
we
ind ing
gene a o s
.
A = x
3
+
y
3
+
z
3 + xz
2
+
y
2
z
+
xy
2+xyz
B
1
=
x
4
+
y4
+
z
4
+
x
2
y
2
+
y
2z 2
+
z 2
x2 +
xyz(x+y±z),
B2 = x
3
(y+z)
+
xyz(Y+z)
+
z
3
(x+y)
+
x2
y 2
+
y a
z
,
B3 =
x
3y
+
x2
z 2
+ xy3
+
xz
3
+
ya
z
+ y
2
z2
,
C
1
=
x5
+
y5
+
z5
+
xyz(xy+yz+zx)
+xy4
+
xz4
+
y4
z
,
C2 =
xy4,+
yz4
+
zx
4
+
x2Y
3
+
y2z3 +
z 2
x3+x2
Yz(Y+z)

,
C
3
=
x
4
y
+
y4
z
+ z
4
x
+
x3y
2
+
y
3
z
2
+
z 3
x2 +
.
xy3(x+Y)
4
2

4
2

4 2

2
4

2
4

24

3
3
3

2
2
2
D
l
=x
y
+y
z
+z
x
+xy
+y
z
+z
x
+xyz(x+y+z
)
+xy
z
,
D
2
= x5y
+
y5
z
+z
5
x
+
x2y
4
+ y
2
z4 +
z 2
x4 +
xy
2(x3+Y
3 ),

'
D 3
=
x
5
y+ysz + z
s
x
+
x2y
2
z2 +
x
s
z
+
x4
y2 + x4yz +
yzs,
1+
3
(1-
2
)(1- 3
xyz(x
3
y
+
yaz +z3
x
+
xy
3
+
yz3
+
zx
3
) ,
x
6
y
+
y6z
+
z
6
x +
x
5
y2 +
y5z2
+
z S
x2 +
x5y(x+y)
x
6
y
+
y6z
+
z6x +
x3
y4 + y3z
4
+
z 3
x4 +
xy
3(x3+y
3
) .