RingsRofiGeneralizednandgStablesInvariantsofRGeneralizediandnStablegInva*
*riantss of Generalized and Stable Invariants
ofoPseudoreflectionsfandPPseudoreflectionsGroupseudoreflectionsoandfPseudorefl*
*ectionPGroupsseudoreflections and Pseudoreflection Groups
FrankFNeumann,rMaraaD.nNeuselkandNLarryeSmithumann,FMararD.aNeuselnandkLarryN*
*Smitheumann, Mara D. Neusel and Larry Smith
AG : Invariantentheorie
mathematisches institut der universit"at g"ottingen
bunsenstrasse 3 - 5
d 37073 g"ottingen
JOURNAL OF ALGEBRA 182 (1996), 85-122
amsaclassificationm:saclassificationm:s1classification3:A50 Invariant Theory
Typeset by LSTEX
summarys:ummarys:ummaryL:et % : G ! GL (n; IF) be a representation of a finit*
*e group G over the
field IF and IF[V ] the space of polynomial functions on V = IFn. We associate*
* to G an
ideal J1 (G) IF[V ] called the ideal of stable invariants of G (or more precis*
*ely %). If
S GL (n; IF) is a set of pseudoreflections we associate to S an ideal I(S) IF*
*[V ] called
the ideal of generalized invariants of S. When G is a pseudoreflection group we*
* investigate
I(S) for various choices of S G and the relation between J1 (G) and I(S). To*
* a
representation % : G ! GL (n; IF) of a finite group, respectively to a set S *
*GL (n; IF)
of pseudoreflections, we also associate a ring __grJ1 (G), respectively __grI(S*
*). We show that
__gr
I(S)is always a polynomial algebra over IF, and whenever %(G) is generated by*
* semisimple
pseudoreflections S that __grJ1~=_grI(S).
Let IFbe a field and V = IFnan n-dimensional vector space over IF. Denote by IF*
*[V ] the
algebra of polynomial functions on V . We regard IF[V ] as a graded algebra ove*
*r IFwith
homogeneous component of degree d, IF[V ]d, the homogeneous polynomials of degr*
*ee d.
If G is a finite group and % : G ! GL (n; IF) is a faithful representation we *
*denote by
IF[V ]G the ring of invariants of G. (As a general reference for invariant theo*
*ry and many
of the notations we use we refer to [11].)
A linear automorphism s : V -! V is called a pseudoreflection if
(i)s 6= 1,
(ii)s has finite order,
(iii)s leaves a codimension one subspace, called the hyperplane of s, fixed.
If % : G ! GL (n; IF) is a faithful representation and %(G) GL(n; IF) i*
*s gener-
ated by pseudoreflections then we refer 1 to G as a pseudoreflection group. If*
* % :
G ! GL (n; IF) is a finite pseudoreflection group and |G| is relatively prime *
*to the char-
acteristic of IF, then by [10], [3], [12]IF[V ]G = IF[f1; : :;:fn] is a polynom*
*ial algebra. The
polynomials f1; : :;:fn 2 IF[V ] form a regular sequence. If the characteristic*
* of IFdivides
|G| (the order of G) this may no longer be true. For example, the Weyl group W *
*(F4) of
the root system of type F4 has a faithful representation over the Galois field *
*of p elements
as a pseudoreflection group. For p = 3 the ring of invariants is not a polynomi*
*al algebra
(see e.g. [11]x 7.4 and x 10.3 example 1, and [17]).
The purpose of this note is to examine finite pseudoreflection groups in a modu*
*lar setting,
i.e., where the order of the group is divisible by the characteristic of the fi*
*eld. We introduce
the ring of generalized invariants associated to a set of pseudoreflections S *
*GL (n; IF)
and the ring of stable invariants associated to a pseudoreflection group and de*
*velop
their properties. In further notes we compute these rings for Coxeter and Sheph*
*ard groups.
We next review our results.
If s is a pseudoreflection and `s 2 IF[V ] a linear form with
ker(`s) = V s= {v 2 V | s(v) = v} ;
then for each f 2 IF[V ]
sf - f = s(f)`s
in IF[V ] for a unique s(f) 2 IF[V ] (see e.g. [3], [4]or [11]x 7.1). Note that
s(f) = 0 () sf = f :
_________________________1
This is an abuse of terminology. Actually it is % that is a pseudoreflection *
*representation. See [11]
x 5.6 example 1 for a group with two quite different pseudoreflection represent*
*ations.
Frank Neumann, Mara D. Neusel and Larry Smith
If S GL (V ) is a collection of pseudoreflections then the ideal of generalize*
*d invariants
I(S) is defined by [4][6]
n *
* o
I(S) = f 2 IF[V ] | deg(f) > 0 and s1. .s.deg(f)(f) = 0 8 s1; : :;:sdeg(f)2*
*;S
where s1. .s.deg(f)denotes the composition of the operators s1; : :;:sdeg(f). (*
*N.b. If
s 2 GL (V ) is a pseudoreflection then `s, and hence also s are well defined up*
* to a nonzero
scalar. Hence I(S) depends only on S and not the choice of `s for s 2 S.) Follo*
*wing Kac
and Peterson [6]we show that I(S) is generated by a regular sequence h1; : :;:h*
*n 2 IF[V ].
If A is a graded connected commutative algebra over a field IFand I A a proper*
* ideal,
then we may associate to the filtration of A
A I I2 . . .Im . . .
by the powers of the ideal I, (we use the convention that I0 = A) the associate*
*d graded
algebra, which is a bigraded algebra defined by
i j
grI(A)m; n-m = Im =Im+1
n
for n; m 2 IN. By totalizing we obtain a positively graded commutative algebr*
*a over
IF containing A=I is a subalgebra, and __grI(A) = IF A=I grI(A) is a graded con*
*nected
commutative algebra over IF.
Starting with the ideal of stable invariants I(S) of S GL (n; IF) we obtain th*
*e ring of
generalized invariants __grI(S)(IF[V ]). The ring of generalized invariants is *
*shown to be
a polynomial algebra.
If S GL (n; IF) is a collection of pseudoreflections the subgroup of GL (n; IF*
*) generated
by S is denoted by G(S). The ideal I(S) of generalized invariants ofiS contain*
*s the
______j
G(S)-invariant polynomials of positive degree, and hence the ideal IF[V ]G tha*
*t they
generate. If G(S) is a finite group of order prime to the characteristic of th*
*e field IF,
then IF[V ]G(S) = IF[f1; : :;:fn] and I(S) = (f1; : :;:fn). Thus our results ma*
*y be seen as
generalizations of the theorem of Shephard-Todd ([10]theorem 5.1, [11]theorem 7*
*.4.1) to
a modular setting.
Actually one would like to associate a ring of generalized invariants to an arb*
*itrary pseudo-
reflection group rather then to a collection of pseudoreflections. We show by e*
*xamples that
the ideal I(S) does not fulfill this goal. If however the pseudoreflection grou*
*p G is gener-
ated by pseudoreflections of order relatively prime to the characteristic of IF*
*this may be
achieved in the following way.
__
Let A be a graded connected algebra over a field IF. Denote by A A the augment*
*ation
ideal of A, i.e. the ideal of A generated by the elements of positive degree. I*
*f G is a group
and ff : G ! Aut (A) a faithful representation of G as a group of algebra auto*
*morphisms
of A, we denote by AG = {a 2 A | ga = a 8 g 2 G}the subalgebra of A left invari*
*ant by
G. The algebra AG is also graded and connected. We define the algebra of coinva*
*riants
2
Rings of Generalized and Stable Invariants
of G by AG := IFAG A, where A is regarded as an AG module via the inclusion and*
* IF
via the augmentation. The algebra of coinvariants may also be viewed as A=I whe*
*re I is
the ideal generated by a 2 AG | deg(a) > 0 . The ideal I is stable under the a*
*ction of G
on A so there is induced an action of G on AG . It is therefore possible to rep*
*eat the above
constructions forming (AG )G ; AGG := (AG )G etc. By iteration we obtain a seq*
*uence of
algebra epimorphisms
A -! AG -! AGG - ! . . .-! AG...G-! . .:.
Denote by Jm the kernel of the map A -! AG m:::!G. These ideals form an ascen*
*ding
chain
(0) = J0 J1 J2 . . .Jm . . .A
S
in A. The ideal J1 := Jm A is called the ideal of stable invariants. It *
*is
characterized by the following conditions:
(i)J1 A is stable under the G-action,
______
(ii)(A=J1 )G ~=IF (i.e. G acts fixedpoint freely on (A=J1 ),
(iii)if I A is an ideal such that (A=I)G ~= IFthen J1 I. The ring __gr*
*J1 (G)is
the ring of stable invariants of G.
If % : G ! GL (n; IF) is a representation of a finite group G then G acts on I*
*F[V ] and there
is the ideal of stable invariants J1 (G) IF[V ]. If G is a pseudoreflection gr*
*oup and |G| is
relatively prime to the characteristic of IFthen Chevalley [3]has shown that IF*
*[V ]G is the
regular_representation of G. In particular (IF[V ]G )G ~= IFand hence J1 (G) =*
* J1(G) =
(IF[V ]G) is the ideal of IF[V ] generated by the G-invariants of positive degr*
*ee. If however
|G| = 0 2 IFthen J1 (G) can be distinctly larger that J1(G). If S GL (n; IF) i*
*s a set
of pseudoreflections we examine the relationship between I(S); J1 (G(S)) and IF*
*[V ]G(S)
and clarify some remarks of Kac and Peterson in [6]x 3.
We have included a rather large number of examples to illustrate the theory and*
* to prevent
false interpertations, and as such, they are an integral part of our work.
The results of this note have applications to p-compact groups [5], which are a*
* homotopical
generalization of a compact Lie group. These investigations grew out of a study*
* of torsion
phenomena in the cohomology of compact Lie groups and their classifying spaces,*
* with
an eye towards their generalization to p-compact groups, in the Topologie-Obers*
*eminar
in G"ottingen in the winter semester of 1993/94. We wish to thank the other me*
*mbers
of the seminar for their active participation. One of us, M.D.N., would like to*
* thank the
Schweizer National Fond for financial support during the preparation of this ma*
*nuscript.
3
Frank Neumann, Mara D. Neusel and Larry Smith
x 1. The Ideal of Stable Invariants
Let A be a positively graded connected algebra over a field IF and G a group of*
* algebra
automorphisms of A. Denote by AG A the subalgebra of G-invariant elements and *
*set
AG := IFAG A, where A is regarded as an AG -module via the inclusion and IF via*
* the
augmentation. AG is called the algebra of coinvariants. The kernel of the quoti*
*ent map
A -! AG is stable under the action of the group G on A and hence AG inherits a *
*G-action
from A. We may therefore iterate the construction of the coinvariants and indu*
*ctively
define AGm for m 2 IN by AGm = (AGm-1 )G . (Of course AG0 is just A and AG1 i*
*s AG .)
Denote by Jm A the kernel of the natural map A -! AGm . The ideals Jm ; m 2 IN*
*form
an ascending chain
(0) = J0 J1 J2 . . .Jm . . .A
S
and we set J1 = Jm , which is called the ideal of stable invariants. Alter*
*natively,
m2IN
the ideals Jm ; m 2 INmay be defined inductively 2 by
ae
Jm = (0)({a 2 A | ga - a 2 J for m = 0
m-1)}for m > 0 :
If B is a graded connected algebra over IF on which G acts, we say that G acts *
*fixed
point freely on B if BG = IF, in otherwords, if the action of G on the homogen*
*eous
component Bm of B of degree m has only 0 2 Bm as a fixed point for all m > 0.*
* The
action of G on A=J1 =: AG1 is fixed point free. The following lemma shows ho*
*w this
property characterizes J1 .
LemmaL1.1emmaL1.1emma:1.1Let A and B be positively graded commutative conn*
*ected algebras
over a field IFon which the group G acts by algebra automorphisms. Assume the a*
*ction
of G on B is fixed point free. The quotient map q : A -! AG1 has the following*
* universal
property: for any algebra homomorphism ' : A -! B commuting with the G-action, *
*there
exists a unique algebra homomorphism "': AG1 -! B making the following diagram
A __q__AG1//
z
'|| zzz
|fflffl"'__zz
B
___
commute. |__|
Let % : G ! GL (n; IF) be a representation of a finite group. The totalizatio*
*n (see [11]
chapter 4) of the algebra of coinvariants IF[V ]G is finite dimensional and gen*
*erated by the
homogeneous component (IF[V ]G )1 of degree 1, which is the image of V *under t*
*he quotient
map IF[V ] -! IF[V ]G . Since the dimensions of the homogeneous components (IF*
*[V ]Gi)1
decrease with i the algebras IF[V ]Gi become isomorphic for i n and the chain *
*of ideals
(0) = J0 J1(IF[V ]) J2(IF[V ]) . . .Jm (IF[V ]) . . .IF[V ]
_________________________2
If X A then (X) denotes the ideal of A generated by X.
4
Rings of Generalized and Stable Invariants
stabalizes after at most n steps.
The totalization of any of the quotient algebras IF[V ]=Ji(G) is finite dimensi*
*onal, so the
radical of the ideal Ji(G) is the maximal ideal of IF[V ]. Hence each of the id*
*eals Ji(G)
IF[V ]; i 2 INis primary.
ExampleE1.2xampleE1.2xample:1.2Let p be an odd prime and IFa field of chara*
*cteristic p. The matrices
S = 10 11 ; T = -10 01 2 GL (2; IF)
generate a dihedral group of order 2p. We recall [11]x 5.6 example 1 that
IF[x; y]D2p ~=IFp[y; (xyp-1 - xp)2]
where x; y 2 V *= Hom (V; IF) is the dual of the canonical basis for IF2. There*
*fore
IF[x; y]D2p ~=_____IF[x;_y]___(y;~(xyp-1=-Ixp)2)F[x]_(x2p)
where x 2 IF[x; y]D2p is the residue class of x. The action of D2p on x; y is g*
*iven by the
transposed matrices so
S(x) = x + y T (x) = -x
S(y) = y T (y) = y :
From these formulae it follows that
2]
(IF[x; y]D2p)D2p ~=IF[x_(x2p)
and hence
IF[x; y]D2pD2p ~=IF[x]_(x2):
The action of D2p on IF[x; y]D2pD2p is fixed point free, so J2 = . .=.J1 = (x2*
*; y)
IF[x; y] is the ideal of stable invariants.
The ideal of stable invariants is an aspect of the modular case as the followin*
*g result shows.
PropositionP1.3ropositionP1.3roposition:1.3Let % : G ! GL (n; IF) be a re*
*presentation of a finite group.
Let H < G be a subgroup_with |H| 2 IFx. Then Jn(G), for n = 1; 2; : :;:1, is ge*
*nerated
by elements from IF[V ]H. In particular, if |G| 2 IFx then J1(G) = J2(G) = . .=*
*.J1 (G).
______
ProofP:roofP:roofB:y definition J1(G) is generated by IF[V ]Gand IF[V ]G *
*IF[V ]H so the
result is established for n = 1. Proceeding inductively we suppose that Jn(G) i*
*s generated
by elements of IF[V ]H of positive degree. Let f 2 Jn+1(G). Then gf - f 2 Jn(G*
*) for all
g 2 G, say
gf - f = Fg; Fg 2 Jn(G) :
Summing these equations (see [11]x 2.4 for the definition of the transfer TrH *
*and the
projection operator ssH ) over all g 2 H gives
X
TrH (f) - |H| . f = Fh ;
h2H
5
Frank Neumann, Mara D. Neusel and Larry Smith
so solving for f we find
(*) f = ssH (f) + F
_______
where ssH (f) 2 IF[V ]H and F 2 Jn(G). If h1; : :;:hm 2 IF[V ]Hgenerate Jn(G),*
* then we
may find f1; : :;:fk 2 Jn+1(G) such that h1; : :;:hm ; f1; : :;:fk generate Jn+*
*1(G). Equa-
tion (*) then shows that h1; : :;:hm_;_ssH f1; : :;:ssH fk also generate Jn+1(G*
*), completing
the induction step and the proof. |__|
CorollaryC1.4orollaryC1.4orollary:1.4Let % : G ! GL (n; IF) be a represen*
*tation of a finite group over
the field IF. Then "
___
Jn(G) J(H) : |__|
HG ; |H|2IFx
For example if G is a group of order paqb and IFis a field of characteristic p,*
* then Jn(G)
is contained in the intersection of J(H) where H ranges over all the q-Sylow su*
*pgroups of
G.
CorollaryC1.5orollaryC1.5orollary:1.5Let % : G ! GL (n; IF) be a represen*
*tation of a finite group.
Suppose |G| 2 IFx. Then J1 (G) is generated by a regular sequence if and only i*
*f %(G) is
generated by pseudoreflections.
______
ProofP:roofP:roofB:y_1.3 J1 (G) = (IF[V ]G), so there is a regular sequenc*
*e f1; : :;:fn 2
IF[V ]G generating (IF[V ]G). We claim that f1; : :;:fn generate IF[V ]G as an *
*algebra. To
prove this let F 2 IF[V ]G have positive degree. We may proceed inductively and*
* suppose
that for all H 2 IF[V ]G with_deg(H)_< deg(F ) that H belongs to the subalgebra*
* generated
by f1; : :;:fn. Since F 2 (IF[V ]G) there are H1; : :;:Hn 2 IF[V ] such that
Xn
F = Hifi:
i=1
Apply ssG to this equation to obtain
Xn
F = ssG (F ) = ssG (Hi)fi:
i=1
The classes ssG (Hi) for i = 1; : :;:n belong to IF[V ]G and have strictly sma*
*ller degree
than F . Therefore the righthand side of the preceding equation belongs to the *
*subalgebra
generated by f1; : :;:fn completing the inductive step.
Since a regular sequence is algebraically_independent the result follows from t*
*he theorem
of Shephard-Todd [10][3][12]. |__|
The following result is useful when discussing Coxeter groups [7].
6
Rings of Generalized and Stable Invariants
CorollaryC1.6orollaryC1.6orollary:1.6Let % : G ! GL (n; IF) be a represen*
*tation of a finite group.
Assume that J1 (G) = (f1; : :;:fn), where f1; : :;:fn 2 IF[V ] is a regular seq*
*uence. If
Qn
H < G is a subgroup with |H| 2 IFx then |H| divides deg(fi).
i=1
_______
ProofP:roofP:roofB:y 1.3 we may find elements h1; : :;:hn 2 IF[V ]Hwhich a*
*re a minimal
generating set for J1 (G). Then
Tot ___IF[V_]__(h= Tot(IF[V ]=J1 (G))
1; : :;:hn)
is finite dimensional, so by the Noether Normalization Theorem ([11] theorem 5.*
*3.3)
h1; : :;:hn 2 IF[V ]H is a system of parameters. Hence by Macaulay's theorem (*
*[11]the-
orem 6.7.7) h1; : :;:hn is a regular_sequence_in IF[V ], and by Solomon's theor*
*em ([11]
theorem 6.8.1) the result follows. |__|
CorollaryC1.7orollaryC1.7orollary:1.7Let % : G ! GL (n; IF) be a represen*
*tation of a finite group
G. Assume J1 (G) = (f1; : :;:fn), where f1; : :;:fn 2 IF[V ] is a regular seque*
*nce. Let
p(k) denote the power of p in the integer k, i.e. k = pp(k)` with (p; `) = 1. *
* If IF has
Qn
characteristic p then |G|=pp(|G|)divides deg(fi).
i=1
ProofP:roofP:roofA:pplyn1.6oto all p0-Sylow subgroups of G, with p0 6= p, *
*and use that
___
l:c:m: |Sylp0(G)|= __|G|_pp(|G|). |__|
For p-groups the stable invariants are also uninteresting as the following resu*
*lt shows.
PropositionP1.8ropositionP1.8roposition:1.8Let A be a positively graded, c*
*onnected, commutative algebra
of finite type over a_field_of characteristic p 6= 0. If P is a finite p-group *
*of automorphisms
of A then J1 (P ) = (A ) A. (Or put another way, AP1 = IF.)
ProofP:roofP:roofI:f M is a nontrivial finite dimensional P -module then t*
*he class equation
(see e.g. [11]pp.85) shows
X
|M| = |MP | + |P : Pi|
i
where the sum ranges over the nontrivial isotropy subgroups of P on M. Hence |M*
*P |
0 mod p so MP 6= {0}.
Therefore if the homogeneous component [APm ]k of degree k of APm is not zero,*
* then
dim ([APm ]k) > dim([APm+1 ]k). Since dim(Ak) is finite it_follows that for lar*
*ge enough m
and any k 2 INthat [APm ]k = 0 and the result follows. |__|
If % : P ! GL(n; IF) is a representation of a finite p-group over a field of *
*characteristic p
then the smallest integer k for which Jk(P ) = (V *) is the socle length of V **
*regarded as
a module over the group ring IF(P ) (see e.g. [2]chapter 1).
7
Frank Neumann, Mara D. Neusel and Larry Smith
PropositionP1.9ropositionP1.9roposition:1.9Let % : G ! GL (n; IF) be a re*
*presentation of a finite group.
Assume that J1 (G) = (f1; : :;:fn) IF[V ] is generated by n elements. Then f1;*
* : :;:fn
is a regular sequence.
ProofP:roofP:roofS:ince IF[V ]=J1 (G) is a quotient of IF[V ]G it follows *
*that Tot(IF[V ]=J1 (G))
is finite dimensional. Hence f1; : :;:fn 2 IF[V ] is a system of parameters by *
*the Noether
Normalization Theorem ([11]theorem 5.3.3). Hence by Macaulay's theorem_([11]the*
*orem
6.7.7) it follows that f1; : :;:fn 2 IF[V ] is a regular sequence. |__|
As with the case of rings of invariants, the stable invariants of the full gene*
*ral linear group
GL (n; IF) over a finite field IF are universal. Here is a computation of thes*
*e universal
stable invariants in a special case.
ExampleE1.10xampleE1.10xample:1.10Consider the group GL (2; IF2). The group*
* GL (2; IF2) has order 6
(abstractly it is isomorphic to 3, the symmetric group on 3 letters) and the el*
*ements are
T = 01 10 A = 10 11 B = 11 01
I = 10 01 T A = 01 11 AT = 11 10 ;
so T; A; B generate GL (2; IF2). The action of GL (2; IF2) is therefore determi*
*ned by the
three involutive pseudoreflections (in this case transvections) T; A; B, and is*
* given by the
schema:
T A B
x 7! y x 7! x + y x 7! x
y 7! x y 7! y y 7! x + y :
The invariants of this group are [11]section 8.1
IF2[x; y]GL (2; IF2)= IF2[d0; d1] ;
where d0; d1 are the Dickson polynomials
d0 = x2y + xy2; deg(d0) = 3
d1 = x2 + xy + y2; deg(d1) = 2:
The coinvariants IF2[x; y]GL (2; IF2)are a Poincare duality algebra (see e.g.[1*
*1]x 6.5) of
dimension 3. One way to visualize this algebra is with the aid of the followin*
*g diagram
(see [11]section 1.2)
x3 + x2y + y3
o
x2 o o y2
x o o y
o
1
Diagram 1.1 :IF[x; y]GL (2; IF2)
8
Rings of Generalized and Stable Invariants
where the nodes on a horizontal level indicate basis elements of degree equal t*
*o the height of
the node above the node labeled 1, which has degree 0. The polynomial h = x3+x2*
*y+y3 2
IF2[x; y]GL (2; IF2)must be invariant under the action of GL (2; IF2) on IF2[x;*
* y]GL (2; IF2)
because the group GL (2; IF2) has no nontrivial 1-dimensional representations o*
*ver IF2.
The following formulas (details left to the reader)
h - T h= d0 2 J1
h - Ah = d0 2 J1
h - Bh = d0 2 J1
indeed verify that h 2 J2.
In fact (d0; d1; h) = J2 = J3 = . .=.J1 . To see this note that the quotient map
IF2[x; y] -! IF2[x; y]GL (2; IF2)
is an isomorphism in degree 1. Hence there no fixed points in IF2[x; y]GL (2; I*
*F2)of degree
1, since there are none in IF2[x; y] of degree 1. The Frobenous map i : ` 7! *
*`2 is a
GL (2; IF2)-equivariant isomorphism
i
IF2[x; y]GL (2; IF2)1-!IF2[x; y]GL (2; IF2)2
and therefore IF2[x; y]GL (2; IF2)2contains no fixed points. This shows that J*
*2 = J3 =
. .=.J1 .
The polynomial h 2 IF2[x; y] is left invariant by the matrix C = T A, and the s*
*ubalgebra
of IF2[x; y] generated by h; d0; d1 is the ring of invariants of the cyclic sub*
*group ZZ=3
GL (2; IF2) generated by the matrix
C = 01 11 2 GL (2; IF2) :
This may be verified by using Molien's theorem to compute the Poincare series of
IF2[x; y]ZZ=3 ([11] section 4.3) and the theorem of Hochster and Eagon ([11] s*
*ection
6.7). The first step is to choose a characteristic zero lift 3 of the represen*
*tation % :
ZZ=3 ! GL (2; IF2). This is accomplished by the representation "%: ZZ=3 ! GL*
* (2; ZZ)
generated by the matrix
"C= 0 -1 2 GL (2; ZZ) :
1 -1
Molien's theorem ([11]theorem 4.2) yields the following formula for the Poincar*
*e series of
_________________________3
The operation of GL(2; IF2) on IF22permutes the nonzero elements of IF22cycl*
*ically. This suggests the
characteristic zero lift is the cyclic permutation of the third complex roots o*
*f unity in the hexagonal lattice
in |C. For another example of this sort see the discussion of the mod 3 represe*
*ntation of the quaternion
group in [11]section 4.3 example 1.
9
Frank Neumann, Mara D. Neusel and Larry Smith
IF2[x; y]ZZ=3:
2 3
6 1 1 1 7
P (IF2[x; y]ZZ=3;=t)1_366______________ + _______________+ _______________77
4det 1 - t 0 1 t 1 + t -t 5
0 1 - t det -t 1 + t det t 1
= 1_3___1___(1+-_t)21____1++_t_+1t2__1 + t + t2
..
.
3
= ____1_+_t_____(1:- t2)(1 - t3)
The ring IF2[x; y]ZZ=3is Cohen-Macaulay and d0; d1 2 IF2[x; y] are a system of *
*parameters.
Hence IF2[x; y]ZZ=2is a free finitely generated module over IF2[d0; d1]. The Po*
*incare series
3
of IF2[d0; d1] IF2[d0; d1] . h is also ___1+t____(1-t2)(1-t3), so the map
IF2[d0; d1] IF2[d0; d1] . h -! IF2[x; y]ZZ=3
is a surjection, and hence an isomorphism. Therefore IF2[x; y]ZZ=3is the subal*
*gebra of
IF2[x; y] generated by the stable invariants d0; d1; h of GL (2; IF2).
The stable invariants of the finite general linear groups are the subject of an*
* ulterior
investigation.
Another important new feature of invariant theory over Galois fields is the Ste*
*enrod algebra
(see e.g. [14], [11]chapters 10 and 11), which operates on the ring of invaria*
*nts. The
existence of these operations impose significant restrictions on the subalgebra*
*s of IFq[V ]
which can be rings of invariants. See e.g. [1], [11]chapter 10, [16], and [19].
PropositionP1.11ropositionP1.11roposition:1.11Let % : G -! GL (n; IFq) be *
*a representation of a finite group
G. Then the ideals Ji(G) IFq[V ] are closed under the action of the Steenrod a*
*lgebra for
i = 1; 2; ; : :;:1. Hence IF[V ]=Ji(G) is an unstable algebras over the Steenro*
*d algebra for
i 2 IN.
___
ProofP:roofP:roofT:his is immediate from the Cartan formula. |__|
x 2. The Ideal of Generalized Invariants
Let s 2 GL (n; IF) be a pseudoreflection and `s 2 IF[V ] a linear form with ker*
*(`s) = Hs =
{v 2 V | sv = v}. The linear form `s depends on s only up to a nonzero scalar. *
*If f 2 IF[V ]
has positive degree then
(s - 1)(f) = `s . s(f)
10
Rings of Generalized and Stable Invariants
for a unique s(f) 2 IF[V ] (see e.g. [11]x 7.1 and the references there). If f *
*has degree k
then s(f) has degree k - 1, and s(f) = 0 if and only if sf = f. For a 2 IF[V ] *
*of degree
0 set s(a) = 0.
The operator s : IF[V ] -! IF[V ] is linear and satisfies the following twisted*
* derivation
formula:
s(fh) = s(f) . h + s(f) . s(h)
(see e.g. [4]). Hence inductively one obtains
i j
s(fk) = s(f) fk-1 + fk-2 . s(f) + . .+.s(f)k-1
for any k 2 IN.
The polynomial s(f) depends on the choice of `s. However, `s is well defined u*
*p to a
nonzero scalar, so s(f) depends on s only up to a nonzero scalar also. If s1; :*
* :;:sm 2
GL (n; IF) are pseudoreflections, and f 2 IF[V ] with deg(f) < m, then s1. .s.m*
*(f) =
0 for degree reasons. Therefore the longest composition s1. .s.mthat can evalu*
*ate
nontrivially on f occurs when m = deg(f). In this case s1. .s.m(f) has degree z*
*ero so
may be identified with a field element.
If S GL (n; IF) is a set of pseudoreflections a polynomial f 2 IF[V ] of posit*
*ive degree is
called an S-generalized invariant if
s1. .s.deg(f)(f) = 0
for all deg(f)-tuples s1; : :;:sdeg(f)2 S. The set of all generalized invariant*
*s of S is denoted
by I(S). Note that I(S) is independent of the choices `s; s 2 S, and hence depe*
*nds only
on S GL (n; IF).
We begin by reviewing some results of Kac and Peterson. Since their work uses *
*other
terminology and notations, and has appeared only in abridged form [6]we include*
* proofs
of our own in the hopes of improving the readibility of this note.
LemmaL2.1emmaL2.1emma(2.1Demazure, Kac and Peterson): Let S GL (n; IF) be*
* a set of pseudo-
reflections. Then I(S) IF[V ] is an ideal.
ProofP:roofP:roofL:et h 2 I(S) have degree k and f 2 IF[V ] have degree l.*
* We show by
double induction on k and l that f . h 2 I(S).
For l = 0 there is nothing to prove. For k = 1 and l arbitrary we have
s1. .s.l+1(f . h) = s1. .s.lsl+1(f) . h + sl+1(f) . sl+1(h)
= s1. .s.l(sl+1(f) . h)
since s(h) = 0 8 s 2 S as h 2 I(S) and deg(h) = 1. But sl+1(f) has degree l - 1*
* and
h 2 I(S), so by induction on l it follows that sl+1(f) . h 2 I(S). Since sl+1(f*
*) . h has
degree l it follows that s1. .s.l(sl+1(f) . h) = 0 as required.
11
Frank Neumann, Mara D. Neusel and Larry Smith
Proceeding inductively over k we have
s1. .s.k+l(f . h) = s1. .s.k+l-1sk+l(f) . h + sk+l(f) . sk+l(h):
Note that sk+l(h) 2 I(S). Since deg(sk+l(h)) = k - 1 it follows from the induc*
*tion
hypothesis that sk+l(f) . sk+l(h) 2 I(S). The polynomial sk+l(f) has degree l -*
* 1 so by
induction on l it follows that sk+l(f) . h 2 I(S). Hence sk+l(f) . h + sk+l(f) *
*. sk+l(h)
belongs to I(S) and has degree k + l - 1. Hence
s1. .s.k+l-1sk+l(f) . h + sk+l(f) . sk+l(h)= 0
___
completing the double induction. |__|
LemmaL2.2emmaL2.2emma:2.2Let S GL (n; IF) be a set of pseudoreflections. *
* Then f 2 I(S) if
and only if s(f) 2 I(S) for all s 2 S.
ProofP:roofP:roofS:uppose s(f) 2 I(S) for all s 2 S. Let d = deg(f) and s1*
*; : :;:sd 2 S.
Then
s1. .s.d(f) = s1. .s.d-1(sd(f)) = 0
*
* ___
as sd(f) 2 I(S) and deg(sd(f)) = d - 1. The converse is equally straightforward*
*. |__|
For a set of pseudoreflections S GL (n; IF) the group generated by S is denote*
*d by G(S).
It is of course a subgroup of GL (n; IF) generated by pseudoreflections. If f 2*
* IF[V ]G(S) has
positive degree, then_s(f)_=_0 for all s 2 S and hence f 2 I(S). Therefore by 2*
*.1I(S)
contains the ideal (IF[V ]G(S)) of IF[V ] generated by the elements of IF[V ]G(*
*S) of positive
degree.
LemmaL2.3emmaL2.3emma:2.3Let S GL (n; IF) be a collection of pseudoreflec*
*tions, then I(S) is
stable under the action of G(S) on IF[V ]. Hence G(S) operates on IF[V ]=I(S).
ProofP:roofP:roofL:et f 2 I(S). Since S generates G(S) it suffices to show*
* that sf 2 I(S)
for all s 2 S. If s 2 S we have
sf = f + `s . s(f) :
By 2.2s(f) 2 I(S) and hence by 2.1`s . s(f) 2 I(S). By_hypothesis_f 2 I(S) so it
follows from the preceding equation that sf 2 I(S). |__|
Observe for f 2 I(S) that s(f) 2 I(S) for all s 2 S. If T S then I(S) I(T ) s*
*o if
S = S0[ S00then I(S) I(S0) \ I(S00). The following lemmas are an aid in comput*
*ing
examples.
LemmaL2.4emmaL2.4emma:2.4Let S GL (n; IF) be a set of pseudoreflections*
*. Suppose that
s; sk 2 S for some integer 1 < k < |s|. Then I(S) = I(S b {sk}).
12
Rings of Generalized and Stable Invariants
ProofP:roofP:roofS:ince S b {sk} S we have I(S) I(S b {sk}). To establis*
*h the reverse
inclusion we note that Hsk = Hs so we may choose `sk = `s. Hence we have
k(f) - f
sk(f) = s________`
s
k(f) - sk-1(f) sk-1(f) - sk-2(f) s(f) - f
= s______________`+ _________________+ . .+.________
s `s `s
= sk-1 sf_-_f_`+ sk-2 sf_-_f_+ . .+.sf_-_f_
s `s `s
= sk-1 . s(f) + sk-2 . s(f) + . .+.s(f)
= (sk-1 + sk-2 + . .+.1)s(f) :
Let f 2 I(S b {sk}). By 2.2 it suffices to show sk(f) 2 I(S). From the preced*
*ing
formula we have
sk(f) = (1 + s + . .+.sk-1)s(f) :
Since s 2 S b {sk} a second application of 2.2shows_s(f)_2 I(S). From 2.3it fol*
*lows
that (1 + s + . .+.sk-1)s(f) 2 I(S) as required. |__|
LemmaL2.5emmaL2.5emma:2.5Let s1; : :;:sk 2 GL (n; IF) be pseudoreflections*
* such that s1 . .s.j
is a pseudoreflection for j = 1; : :;:k. If `si 2 IF[V ]; i = 1; : :;:k are li*
*near forms with
ker(`si) = Hsithen
Xk i ji j
s1...sk(f) = __1___` s1 . .s.i-1(`si) s1 . .s.i-1(si(f))
is1...sk=1
for all f 2 IF[V ].
ProofP:roofP:roofB:y induction on k. For k = 2 we have
i j 1 i j
s1s2(f) = __1__`(s1s2)(f) - f = _____ (s1s2)(f) - s1(f) + s1(f) - f
s1s2 `s1s2
i j
= __1__`s1(`s2)s1(s2(f)) + `s1s1(f) :
s1s2
For k > 3 we obtain from the inductive hypothesis
s1...sk(f) = s1...sk-1.sk(f)
i ji j
= ____1_____` `s1...sk-1s1...sk-1(f) + s1 . .s.k-1(`sk) s1 . .s.k-1(sk(f))
s1...sk-1.sk
k-1Pi ji ji ji *
* j
= __1___` s1...si-1(`si) s1...si-1(si(f))+ s1...sk-1(`sk) s1...s*
*k-1(sk(f))
s1...ski=1
Xk i ji j
= ___1___` s1 . .s.i-1(`si) s1 . .s.i-1(si(f)) ;
s1;:::;isk=1
___
and the result follows. |__|
13
Frank Neumann, Mara D. Neusel and Larry Smith
LemmaL2.6emmaL2.6emma:2.6Let S0; S00 GL (n; IF) be sets of pseudoreflectio*
*ns with G(S0) =
G(S00). Suppose that G(S0) b {1} consists only of pseudoreflections. Set S = *
*S0 [ S00.
Then I(S) = I(S0) \ I(S00).
ProofP:roofP:roofT:he inclusion I(S) I(S0) \ I(S00) is elementary. Let f *
*2 I(S0) \ I(S00).
We prove f 2 I(S) by induction over deg(f) thereby establishing the reverse inc*
*lusion
I(S) I(S0) \ I(S00), and hence the lemma.
If deg(f) = 1 then f 2 I(S) if and only if s(f) = 0 for all s 2 S. This means h*
*owever
that s(f) = 0 for all s 2 S0 and for all s 2 S00, i.e., f 2 I(S0) \ I(S00).
If deg(f) > 1 then
s01...s0deg(f)(f)= 0 8 s01; : :;:s0deg(f)2 S0
s001...s00deg(f)(f)= 0 8 s001; : :;:s00deg(f)2 S00
and we must show
s1...sdeg(f)(f) = 0 8 s1; : :;:sdeg(f)2 S = S0[ S00:
Since sdeg(f)2 G(S0) = G(S00) we have formulas
ae 0 0 0 0 0
sdeg(f)= t1t.0.t.kfor0t1;0:0:;:tk020S0000
1 . .t.lfor t1; : :;:tl 2 S .
Therefore by lemma82.5we have
>>> P k 0 0 0 0 0*
* 0 0
< t01...t0k(f) = i=1 t1. .t.i-1(`t0i) t1. .t.i-1(t0i(f))for t1*
*; : :;:tk 2 S
sdeg(f)(f) = >
>>:t00 (00f) = P l t00. .t.00(` 00) t00. .t.00( 00(f)0for t00*
*; : :;:t002 S00
1...tl i=1 1 i-1 ti 1 i-1 ti 1 *
* l
which yields ae
0)
sdeg(f)(f) 22I(SI(S00) ;
which implies s(f) 2 I(S0)\I(S00) for all s 2 S0[S00._Applying_the induction hy*
*pothesis
then yields f 2 I(S) completing the inductive step. |__|
For a discussion of the subgroups G of GL (n; IF) such that every nonidentity e*
*lement is a
pseudoreflection see [11]chapter 8 section 2, 8.2.1, 8.2.11 and 8.2.17.
The initial goal was to associate to a pseudoreflection group % : G ! GL (n; I*
*F) an ideal
of generalized invariants. Associated to G are however several different natur*
*al sets of
pseudoreflections, such as, in the notations of [11]chapter 7 for example:
s(G) = {s 2 G | %(s) is a pseudoreflection}
s (G)= {s 2 G | %(s) is a diagonalizable pseudoreflection}
s6 (G)= {s 2 G | %(s) is a transvection} :
For a Coxeter group there is also the set of defining generating reflections. T*
*hese sets can
have different ideals of generalized invariants, whose significance depends on *
*the situation,
as the following examples show. (See however 3.4.)
14
Rings of Generalized and Stable Invariants
ExampleE2.7xampleE2.7xample:2.7Consider the matrices
S = 10 11 ; T = -10 01 2 GL (2; IF)
where IFis a field of characteristic p 6= 0; 2. Direct computation shows that
Sp = I = T 2; T ST -1= S-1
so that the subgroup of GL (2; IF) generated by S and T is a dihedral group D2p*
* of order
2p. One has
s(D2p)= {Si; T; T Si | i = 1; : :;:p - 1}
s (D2p)= {T; T Si | i = 1; : :;:p - 1}
s6 (D2p)= {Si | i = 1; : :;:p - 1} :
The sets s(D2p) and s (D2p) both generate D2p. By lemma 2.4
I(s(D2p))= I({S; T; T Si})
I(s6 (D2p)= I({S}) :
Denote by x; y 2 V * IF[V ] the linear forms dual to the standard basis. The*
* fixed
hyperplanes of the pseudoreflections in D2p are
ae oe
HSi = b0 2 IF2 = ker(y)
ae oe
HT = 0b 2 IF2 = ker(x)
ae oe
HTSi = -ia2a 2 IF2 = ker(2x + iy) :
This gives the following formulae
if - f
Si(f) = S______y
T (f) = T_f_-_f_x
if - f
TSi(f) = T_S______2x:+ iy
The action of the twisted derivations on x and y are given by
Si(x) = i Si(y) = 0
T (x) = -2 T (y) = 0
TSi(x) = -1 TSi(y) = 0
and therefore y belongs to all three of the ideals I(s(D2p)); I(s (D2p)) and I*
*(s6 (D2p)).
15
Frank Neumann, Mara D. Neusel and Larry Smith
Next note that ae
S . .S.(xi) = i! 6= 0for i < p
i ! 0 for i = p
so that xp 2 I({S}) = I(s6 (D2p)), and therefore
I(s6 (D2p)) = (y; xp) :
Likewise the formula
TSi(x2) = iy
implies
SjTSi(x2) = Sj(iy) = 0
T TSi(x2) = T (iy) = 0
TSjTSi(x2) = TSj(iy) = 0
so x2 2 I(s (D2p)). Hence
I(s (D2p)) = (y; x2) :
Finally since I(s(D2p)) I(s (D2p)) \ I(s6 (D2p)) it follows that
I(s(D2p)) = (y; x2p) :
Hence the ideals I(s(D2p)); I(s (D2p)) and I(s6 (D2p)) are all distinct.
ExampleE2.8xampleE2.8xample:2.8Let p be an odd prime and IF a field of char*
*acteristic p. Consider
the matrices
S = 10 11 ; T = -10 01 ; D = -10 -11 2 GL (2; IF) :
From example 2.7we have that the subgroup generated by S and T is the dihedral *
*group
D2p of order 2p. Since D = T S it follows that D2p is also generated by T and *
*D. The
matrices S and T are pseudoreflections of orders p and 2 respectively. Direct c*
*omputation
shows D2 = I and hence the matrix D is a pseudoreflection of order 2.
Set S = {S; T }; D = {T; D}. Then S; D GL (2; IF) are sets of pseudoreflection*
*s and
G(S) = D2p = G(D).
Denote by x; y 2 V * IF[V ] the linear forms dual to the standard basis of V = *
*IF2. The
fixed hyperplanes of S; T and D are
ae oe
HS = b0 2 IF2 = ker(y)
ae oe
HT = 0b 2 IF2 = ker(x)
ae oe
HD = -a2a 2 IF2 = ker(2x + y) :
16
Rings of Generalized and Stable Invariants
This gives the following formulae
S(f) = Sf_-_f_y
T (f) = T_f_-_f_x
D (f) = Df_-_f_2x:+ y
The action of the twisted derivations on x and y are given by
S(x) = 1 S(y) = 0
T (x) = -2 T (y) = 0
D (x) = -1 D (y) = 0
so as in example 2.7we have
ae
S . .S.(xi) = i! 6= 0for i < p
i ! 0 for i = p
and xp 2 I({S}) is the lowest power of x in I(S) so I(S) = (y; xp).
By contrast
T (x2) = (x + T (x))T (x) = (x - x)(-2) = 0
D (x2) = (x + D(x))D (x) = (x - x - y)(-1) = y :
Hence
T T (x2) = 0
T D (x2) = T (y) = 0
D T (x2) = 0
D D (x2) = D (y) = 0 ;
and therefore x2 2 I(D). Since x 6= I(D) it therefore follows that I(D) = (y; x*
*2). Hence
I(S) 6= I(D).
These examples show that for a collection T GL (n; IF) of pseudoreflections, t*
*he ideal
of generalized invariants I(T ) is not an invariant of G(T ). From section 1exa*
*mple 1.2we
have that J1 (D2p) = (x2; y) = I(D). This is a special case of the more general*
* result 3.4
below. The ideals of generalized invariants computed so far are in each case ge*
*nerated by
a regular sequence of maximal length. This too is no accident as we show next.
TheoremT2.9heoremT2.9heorem(2.9Kac and Peterson): Let S GL (n; IF) be a c*
*ollection of pseudo-
reflections. Then I(S) is generated by a regular sequence of length n.
The proof of this theorem rests on the following result of Vasconcelos ([18]The*
*orem 1.1),
which we restate in the form we require.
17
Frank Neumann, Mara D. Neusel and Larry Smith
TheoremT2.10heoremT2.10heorem(2.10Vasconcelos): Let A be a graded connecte*
*d commutative algebra
over a field IF and I A a homogeneous ideal of finite projective dimension._Th*
*en_I is
generated by a regular sequence if and only if I=I2 is a free A=I-module. |__|
ProofPofrTheoremo2.9o:fPofrTheoremo2.9o:fBofyTheoremH2.9i:lbert's syzygy t*
*heorem ([11]theorem 6.3.1) the
algebra IF[V ] has finite global dimension. Therefore by the theorem of Vasconc*
*elos 2.10
I := I(S) is generated by a regular sequence if and only if I=I2 is a free IF[V*
* ]=I-module.
Choose an IF-vector space basis f1; : :;:fm for the module QI = IFIF[VI]of inde*
*com-
posables of I. Let f1; : :;:fm 2 I lift f1; : :;:fm to a minimal ideal basis fo*
*r I. Denote by
(IF[V ]=I) . F the free IF[V ]=I-module with generator F . Define a map
' : mi=1(IF[V ]=I) . Fi- ! I=I2
where deg(Fi) = deg(fi) for i = 1; : :;:m in the following way: for*
* an element
m
(h1F1; : :;:hm Fm ) 2 (IF[V ]=I) . Fi choose h1; : :;:hm 2 IF[V*
* ] lifting
i=1
h1; : :;:hm 2 IF[V ]=I and set
'(h1F1; : :;:hm Fm ) = h1f1 + . .h.mfm ;
where the righthand side is to be interpreted as an element of the quotient I=I*
*2 of I.
We claim that ' is an isomorphism. To see this note first that ' is an epimorph*
*ism because
f1; : :;:fm 2 I generate I as an IF[V ]-module, and ' is a homomorphism of IF[V*
* ]-modules.
It remains to show that ' is a monomorphism.
Suppose that h1F1 + . .h.mFm 2 ker('). We need to show that hi = 0 2 IF[V ]=I*
* for
i = 1; : :;:m. Let us suppose to the contrary. By reordering we may also suppose
deg(h1) . . .deg(hm ) ;
and that j 2 {1; : :;:m} is chosen so that hi= 0 for i < j, while hj 6= 0. Set *
*deg(hj) = d.
Choose polynomials hi 2 IF[V ] lifting hi 2 IF[V ]=I for i = 1; : :;:m. Withou*
*t loss of
generality we may suppose that hi = 0 for i < j. Since hj 62 I there are pseudo*
*reflections
s1; : :;:sd 2 S such that
(s1. .s.d)(hj) 6= 0 2 IF:
The ideals I and I2 are invariant under the operators {s | s 2 S} and hence s a*
*cts also
on IF[V ]=I and I=I2 for any s 2 S. Apply s1. .s.mto the equation
'(hjFj + . .h.mFm ) = 0
to obtain
s1. .s.d(hjfj + . .+.hm fm ) = 0
in I=I2. By iterating the twisted derivation formula for s(f . h) we obtain
X X
0 = s1. .s.d(hi) . fi+ ui. wi;
deg(hi)=d
18
Rings of Generalized and Stable Invariants
where ui 2 IF[V ]=I and wi 2 I=I2 with deg(ui); deg(wi) > 0. Let q : I=I2 -! Q*
*I be the
quotient map. Apply q to the preceding equation to obtain
X
s1. .s.d(hi) . fi= 0 2 QI :
deg(hi)=d
By hypothesis s1. .s.d(hj) 6= 0 and deg(hj) = d. Therefore the preceding equati*
*on is
a nontrivial linear relation between f1; : :;:fm contrary to their choice as an*
* IF-basis for
QI. Hence ' is also a monomorphism.
Let__I(S)_ = (h1; : :;:hm ) where h1; : :;:hm 2 IF[V ] is a regular se*
*quence. Since
(IF[V ]G(S)) I(S) and IF[V ] is finite over IF[V ]G(S) it follows that IF[V *
*]=I(S) =
IF[V ]=(h1; : :;:hm ) is finite dimensional. Hence h1; : :;:hm 2 IF[V ] is al*
*so a system of
parameters,_and therefore by Macaulay's theorem ([11]theorem 6.7.11) we conclud*
*e that
m = n. |__|
Thus for any set of pseudoreflections S in GL (n; IF) the ideal of generalized *
*invariants
I(S) IF[V ] is generated by a regular sequence of length n = dimIF(V ). We see*
*k an analog
of the theorem of Shephard-Todd et al. ([11]5.5.5) relating the degrees of the *
*polynomials
in such a regular sequence to S.
LemmaL2.11emmaL2.11emma:2.11Let H GL (n; IF) be a subgroup with |H| 2 IFx*
*. Let h1; : :;:hn 2
IF[V ] be a regular sequence such that the ideal (h1; : :;:hn) IF[V ] isPstabl*
*e under the
action of H. Let ssH : IF[V ] -! IF[V ]H be the averaging operator _1_|H| h. Th*
*en
h2H
(i)ssH (h1); : :;:ssH (hn) 2 IF[V ] is a regular sequence,
(ii)ssH (h1); : :;:ssH (hn) 2 IF[V ]H is a regular sequence, and
(iii)(ssH (h1); : :;:ssH (hn)) = (h1; : :;:hn) IF[V ].
ProofP:roofP:roofS:ince (h1; : :;:hn) IF[V ] is stable under the action o*
*f H and the averaging
map is an epimorphism of IF[V ]H -modules we obtain an epimorphism
ssH : IF[V ]=(h1; : :;:hn) -! IF[V ]H =(ssH (h1); : :;:ssH (hn)) :
Since h1; : :;:hn 2 IF[V ] is a regular sequence IF[V ]=(h1; : :;:hn) is finite*
* dimensional, and
therefore so is IF[V ]=(ssH (h1); : :;:ssH (hn)). The ring IF[V ]H has Krull d*
*imension n and
therefore ssH (h1); : :;:ssH (hn) 2 IF[V ]H is a system of parameters. Hence w*
*e have finite
extensions
IF[ssH (h1); : :;:ssH (hn)] IF[V ]H IF[V ]
and therefore ssH (h1); : :;:ssH (hn)IF[V ] is a system of parameters. By Macau*
*lay's theorem
([11]theorem 6.7.7) ssH (h1); : :;:ssH (hn) 2 IF[V ] is a regular sequence.
The averaging operator ssH is an IF[V ]H -module splitting, and hence *
* a fortiori an
IF[ssH (h1); : :;:ssH (hn)]-module splitting, to the inclusion IF[V ]H ! IF[*
*V ] and hence
IF[V ]H is a projective, and therefore free IF[ssH (h1); : :;:ssH (hn)]-module*
* (see e.g. [11]
section 6.1), so by Macaulay's theorem ([11]theorem 6.7.11) ssH (h1); : :;:ssH *
*(hn) 2 IF[V ]H
is a regular sequence.
19
Frank Neumann, Mara D. Neusel and Larry Smith
The classes ssH (h1); : :;:ssH (hn) belong to (h1; : :;:hn) so we have an epimo*
*rphism
q : IF[V ]=(ssH (h1); : :;:ssH (hn)) -! IF[V ]=(h1; : :;:hn) :
The Poincare series computation
Yn "1 - tdeg(ssH (hi))#
P (IF[V ]=(ssH (h1); : :;:ssH (hn));=t) ______________
i=1 1 - t
Yn "1 - tdeg(hi)#
= __________ = P (IF[V ]=(h1; : :;:hn);*
* t)
i=1 1 - t
*
* ___
then implies that q is an isomorphism, so (ssH (h1); : :;:ssH (hn)) = (h1; : :;*
*:hn). |__|
LemmaL2.12emmaL2.12emma:2.12Let H GL (n; IF) be a subgroup with |H| 2 IFx*
*. Let h1; : :;:hn 2
IF[V ] be a regular sequence such that the ideal (h1; : :;:hn) IF[V ] is stabl*
*e under the
Qn
action of H. Then |H| divides deg(hi).
i=1
ProofP:roofP:roofB:y 2.11 we may suppose without loss of generality that h*
*1; : :;:hn 2
IF[V ]H . Since h1; : :;:hn 2 IF[V ] is a regular sequence it follows that IF[*
*V ] is a free
IF[h1; : :;:hn]-module. The averaging map
ssH : IF[V ] -! IF[V ]H
is an IF[V ]H -module splitting of the inclusion IF[V ]H IF[V ]. Therefore *
*IF[V ]H is
an IF[h1; : :;:hn]-module direct summand of IF[V ], and hence IF[V ]H is also*
* a free
IF[h1; : :;:hn]-module (see for example [11]section 6.1). Therefore
H
IF[V ]H ~= IF[h1; : :;:hn] __IF[V_]___(h:
1; : :;:hn)
Taking Poincare series gives
_1_
__|H|__+ higher terms
(1 - t)n
= P (IF[V ]H ; t) = P (IF[h1; : :;:hn]; t) . P (IF[V ]=(h1; : :*
*;:hn); t)
" ______1______ #
= _deg(h1)...deg(hn)_(1+-ht)nigher.termsP (IF[V ]=(h1; : :;:hn*
*); t)
Multiplying both sides by (1 - t)n and evaluating at t = 1 we obtain
_1__= ________1________ . dim (Tot(P (IF[V ]=(h ; : :;:h )))
|H| deg(h1) . .d.eg(hn) IF 1 n
___
and the result follows. |__|
20
Rings of Generalized and Stable Invariants
PropositionP2.13ropositionP2.13roposition(2.13Kac-Peterson): Let S GL (n;*
* IF) be a set of pseudoreflections
and H G(S) a subgroup with |H| 2 IFx. If h1; : :;:hn 2 IF[V ] is a regular se*
*quence
Qn
generating the ideal of generalized invariants I(S) then |H| divides deg(hi).
i=1
___ ProofP:roofP:roofI:(S) is stable under the action of H by 2.3so the result*
* follows from 2.12.
|__|
CorollaryC2.14orollaryC2.14orollary(2.14Kac-Peterson): Suppose that IFis a*
* field of characteristic p. Let
S GL (n; IF) be a set of pseudoreflections and h1; : :;:hn 2 IF[V ] a regular *
*sequence
Qn *
* |G|
generating the ideal of generalized invariants I(S). Then deg(hi) is divisib*
*le by ______p(|G|),
i=1 *
* p
where p(k) denotes the power of p in k.
ProofP:roofP:roofF:or each prime p0different from the characteristic of IF*
*the order |Sylp0(G(S))|
Qn
of the p0-Sylow subgroup Sylp0(G(S))| of G(S) divides deg(hi) by 2.13. Hence*
* the great-
i=1
Qn
est common divisor of these numbers divides deg(hi) by 2.13. But the greates*
*t common
n o i=1 ___
divisor of |Sylp0(G(S))| | p06= pis __|G|_pp(|G|). |__|
When IF is a Galois field the ideal of generalized invariants relates well to t*
*he Steenrod
operations. (See [11]x 10.2 - x 10.6 for a discussion of Steenrod operations a*
*nd their
relevance to invariant theory.)
PropositionP2.15ropositionP2.15roposition:2.15Let s 2 GL (n; IFq) be a pse*
*udoreflection, `s 2 IFq[V ] a linear
form with ker(`s) = Hs and s : IFq[V ] -! IFq[V ] an associated twisted derivat*
*ion. Define
the algebra homomorphism 4
P () : IFq[V ] -! IFq[V ][[]]
by the requirement
1X
P ()(f) = Pi(f)i:
i=0
Then 5
s(P ()(f)) = P () (s(f)) (1 + `q-1s) ;
P P
where s is extended to IFq[V ][[]] by s( fii) = s(fi)i.
_________________________4
5 If A is a ring then A[[]] denotes the ring of formal power series over A in t*
*he variable .
As usual for q = 2 replace Pi(f) by Sqi(f).) To consider P() as a ring homomo*
*rphism of degree 0 we
give the degree (1-q) and the elements of V the degree 1. (Since sign conventi*
*ons for commutation rules
play no role in what follows we employ the grading conventions preferred by alg*
*ebraists and not those of
topologists.)
21
Frank Neumann, Mara D. Neusel and Larry Smith
ProofP:roofP:roofT:he group GL (n; IFq) acts on IFq[V ] and on IFq[V ][[]]*
* by acting trivially
on . The map P () : IFq[V ] -! IFq[V ][[]] is GL (n; IFq) equivariant and an *
*algebra
homomorphism. Therefore on the one hand
P ()(`ss(f)) = P ()(sf - f) = P ()(sf) - P ()(f)
= s(P ()(f)) - P ()(f) ;
while on the other hand
i ji j
P ()(`ss(f)) = P ()(`s) P ()(s(f))
= (`s + `qs)P ()(s(f)) :
Equating gives
s(P ()(f)) - P ()(f) = (`s + `qs)P ()(s(f)) :
Dividing by `s yields
s(P ()(f)) = s(P_()(f))_-_P_()(f)_`= (1 + `q-1s)P ()(s(f))
s
___
as required. |__|
CorollaryC2.16orollaryC2.16orollary:2.16Let S GL (n; IFq) be a collection*
*_of_pseudoreflections. Then
the ideal I(S) IFq[V ] is closed under the action of the Steenrod algebra. |*
*__|
x 3. The Relation Between Stable and Generalized Invariants
Let S GL (n; IF) be a set of pseudoreflections. In this section we investigate*
* the relations
between the ideals of generalized invariants I(S), the ring of invariants IF[V *
*]G(S), and the
ideal of stable invariants J1 (G(S)).
LemmaL3.1emmaL3.1emma:3.1Let S GL (n; IF) be a set of pseudoreflections a*
*nd f 2 IF[V ]. Then
f 2 IF[V ]G(S) if and only s(f) = 0 for all s 2 S.
ProofP:roofP:roofT:his follows from the fact that f is invariant under the*
*_cyclic subgroup
generated by s if and only if s(f) = 0 and the fact that S generates G(S). |__|
LemmaL3.2emmaL3.2emma:3.2Let S GL (n; IF) be a set of pseudoreflections.*
* Then I(S)
J1 (G(S)). More specifically, if f 2 I(S) has degree d then f 2 Jd(G(S)).
ProofP:roofP:roofC:hoose f 2 I(S) and set d = deg(f). Then
s1. .s.d(f) = 0 8 s1; : :;:sd 2 G(S) :
We claim that f belongs to Jd(G(S)). To verify this we proceed by induction on *
*d = deg(f).
_____G(S)
If deg(f) = 1 and f 2 I(S) then s(f) = 0 for all s 2 S so by 3.1 f 2 IF[V ]
22
Rings of Generalized and Stable Invariants
J1(G(S)). If we assume d > 1 and the result established for all h 2 I with deg(*
*h) < d,
then rewriting a bit gives
0 = s1. .s.d(f) = (s1. .s.d-1) sd(f) ;
so by the inductive hypothesis sd(f) 2 Jd-1(G(S)). Hence
sf - f = `ss(f) 2 Jd-1(G(S))
for all s 2 S. Since S generates G(S) it follows that
gf - f 2 Jd-1(G(S)) 8 g 2 G(S)
___
so f 2 Jd(G(S)) by definition. |__|
LemmaL3.3emmaL3.3emma:3.3Let S GL (n; IF) be a set of semisimple_pseudore*
*flections,_i.e. each
s 2 S has order relatively prime to the characteristic of IF. Then (IF[V ]=I(S*
*))G(S) =
J1 (G(S))=I(S).
ProofP:roofP:roofC:onsider the map
q : J1 (G(S)) -! IF[V ]=I(S) :
i__________jG(S)
If q(f) 2 IF[V ]=I(S) then
gf - f 2 I(S) 8 g 2 G(S) :
By lemma 3.2I(S) J1 (G(S)) so
gf - f 2 J1 (G(S)) 8 g 2 G(S)
and hence there is an m 2 INwith
gf - f 2 Jm (G(S)) 8 g 2 G(S) :
i_______*
*___jG(S)
But this says that f 2 Jm+1 (G(S)) J1 (G(S)) so q(J1 (G(S))) contains IF[V ]=*
*I(S)
in positive degrees.
i__________jG(S)
To prove the reverse inclusion we show that q(Jm (G(S))) IF[V ]=I(S) by in*
*duc-
______
tion on m. By definition J1(G(S)) = (IF[V ]G) IF[V ]. Since IF[V ]G I(S) it*
* follows
J1(G(S)) I(S) beginning the induction. Assume next that f 2 Jm (G(S)). Then
gf - f 2 Jm-1 (G(S)) and so by the induction hypothesis
q(f - gf) 2 (IF[V ]=I(S))G:
Hence
f - gf - h(f - gf) 2 I(S) 8 g; h 2 G
23
Frank Neumann, Mara D. Neusel and Larry Smith
Let s 2 S have order k. We have
(f - sf) - s(f - sf) 2 I(S) (g = s; h = s)
(f - s2f) - s(f - s2f) 2 I(S) (g = s2; h = s)
..
.
(f - sk-1f) - s(f - sk-1f) 2 I(S) (g = sk-1; h = s)
and adding gives
kf - ksf = k(f - sf) 2 I(S) :
Since k is invertible in IF it follows f - sf 2 I(S) for all s 2 S. As S genera*
*tes G(S) it
follows that f - gf 2 I(S) for_all g 2 G, i.e. Jm (G(S)) I(S) completing the i*
*nduction
step and hence the proof. |__|
TheoremT3.4heoremT3.4heorem:3.4Let S GL (n; IF) be a set of semisimple ps*
*eudoreflections. Then
I(S) = J1 (G(S)).
i__________jG(S)
ProofP:roofP:roofB:y 3.3 IF[V ]=I(S) = J1 (G(S))=I(S). Let ' 2 (IF[V *
*]=I(S))G(S)
with deg(') > 0. Choose f 2 IF[V ] with q(f) = ', where q : IF[V ] -! IF[V ]=I(*
*S) is the
quotient map. For s 2 S let ~~ denote the cyclic subgroup of G(S) generated b*
*y s.
Since the order of s is invertible in IFthe projection operator
|s|X
ss~~~~ = _1_|s|si
i=1
is defined. From the invariance of ' it follows that ss~~~~ (') = '. This lift*
*s to IF[V ] to
give
(*) f - ss~~~~ (f) 2 I(S)
Since ss~~~~ (f) 2 IF[V ]~~~~ , i.e. ss~~~~ (f) is ~~~~-invariant, it follows that *
*s(ss~~~~ (f)) = 0.
Applying s to the equation (*) and applying lemma 2.2we obtain s(f) 2 I(S). Sin*
*ce ___
this holds for all s 2 S it follows from lemma 2.2that f 2 I(S) and hence that *
*' = 0. |__|
ExampleE3.5xampleE3.5xample:3.5Let p be an odd prime and IFa field of chara*
*cteristic p. The matrix
S = 11 01 2 GL (2; IF)
is a (nondiagonalizable) pseudoreflection of order p, so affords a representati*
*on of ZZ=p. By
[11]x 5.6 example 3
IF[x; y]ZZ=p= IF[x; y(yp-1 - xp-1)] :
By proposition 1.8the ideal of stable invariants J1 (ZZ=p) is (x; y).
24
Rings of Generalized and Stable Invariants
To compute the ideal I(s(ZZ=p)) of generalized invariants we may use 2.4to repl*
*ace s(ZZ=p)
by {S}. Note that HS = SpanIF 01 IF2= V . The linear form
x = `S : V -! IF `s uv = u
has kernel Hs and therefore
S(f) = Sf_-_f_x
for any f 2 IF[V ]. The action of S on V *is given by
S(x) = x S(y) = x + y :
Hence
S(x) = S(x)_-_x_x= 0
S(y) = S(y)_-_y_x= x_+_y_-_y_x= 1 :
Hence y 62 I(s(ZZ=p)) so I(s(ZZ=p)) 6= J1 (ZZ=p). The twisted derivation formu*
*la for S
yields i j
S(yk) = S(y) yk-1 + yk-2S(y) + . .+.S(y)k-1
= yk-1 + yk-2(x + y) + . .+.(y + x)k-1 ;
from which it follows that
ae
s. .s.(yi) = i! 6= 0for i < p
i ! 0 for i = p.
Therefore I(s(ZZ=p)) = I({S}) = (x; yp) 6= (x; y) = J1 (ZZ=p). Hence theorem 3.*
*4does
not hold if the elements of S are not of order relatively prime to the characte*
*ristic of
the ground field. In particular, it does not hold for the mod 2 reduction of th*
*e defining
representation of a crystallographic group, i.e. a Weyl group. (See also the di*
*scussion of
the dihedral group in examples 2.7and 2.8.)
CorollaryC3.6orollaryC3.6orollary:3.6Let S GL (n; IF) be a set of semisi*
*mple pseudoreflections.
Then I(S) = (h1; : :;:hn) = J1 (G(S)) where h1; : :;:hn 2 IF[V ] is a regular s*
*equence and
Qn
moreover __|G|_pp(|G|)divides deg(hi).
i=1
___
ProofP:roofP:roofT:his follows from 2.9, 3.4and 1.6. |__|
CorollaryC3.7orollaryC3.7orollary:3.7Let % : G ! GL (n; IF) be a represen*
*tation of a finite group which
may be generated by pseudoreflections of order relatively prime to the characte*
*ristic of IF.
If_S0; S00 G are sets of semisimple pseudoreflections generating G then I(S0) =*
* I(S00).
|__|
25
Frank Neumann, Mara D. Neusel and Larry Smith
CorollaryC3.8orollaryC3.8orollary:3.8Let % : G ! GL (n; IF) be a represen*
*tation of a finite group which
may be generated by pseudoreflections of order relatively prime to the characte*
*ristic of IF.
Let h1; : :;:hn 2 J1 (G) be a regular sequence generating the ideal of stable i*
*nvariants and
let d = max {deg(h1); : :;:deg(hn)}. Then Jd(G) = . .=.J1 (G).
ProofP:roofP:roofB:y theorem 3.4J1 (G) = I(S) for any set S G of pseudore*
*flections of
orders prime to the characteristic_of IFwhich generate G. By 3.2h1; : :;:hn 2 J*
*d(G(S))
and the result follows. |__|
CorollaryC3.9orollaryC3.9orollary:3.9Suppose that IF is a field of charact*
*eristic not equal to 2 and
G ! GL (n; IF) a representation of a finite group G as a Coxeter group. Let S *
* G be
any subset of reflections of order 2 generating G. Then
(i)I(S) = J1 (G)
__________
(ii)G acts fixed point freely on IF[V ]=I(S).
(iii)I(S) = (h1; : :;:hn) = J1 (G(S)) where h1; : :;:hn 2 IF[V ] is a regul*
*ar sequence
and
Qn ___
(iv) _|G|__pp(|G|)divides deg(hi). |__|
i=1
x 4. Rings of Stable and Generalized Invariants
Let S GL (n; IF) be a set of pseudoreflections and I(S) the ideal of generaliz*
*ed invariants
of S. By 2.9I(S) is generated by a regular sequence h1; : :;:hn 2 IF[V ]. It is*
* not clear
what if any relation there is between the subalgebra IF[h1; : :;:hn] of IF[V ] *
*generated
by h1; : :;:hn and IF[V ]G(S), or even if there is any invariant meaning to be *
*assigned
to IF[h1; : :;:hn]. In this section we show how to associate to S a ring of ge*
*neralized
invariants, and a ring of stable invariants that have invariant meaning.
ConstructionC:onstructionC:onstructionL:et A be a graded connected commutat*
*ive algebra over a field IF
and I A a proper ideal. Introduce the filtration of A
A I I2 . . .Im . . .
by the powers of the ideal I. Denote by grI(A) the associated graded algebra. T*
*his is a
bigraded algebra with i j
grI(A)m; n-m = Im =Im+1
n
for n; m 2 IN, where we have used the convention that I0 = A. The algebra grI(A*
*) may
be totalized to yield a positively graded commutative algebra over IF. In the s*
*equel grI(A)
denotes this totalization. Note that A=I grI(A) is a subalgebra, so we may re*
*gard
grI(A) as an algebra over A=I. Finally we set __grI(A) = IFA=I grI(A), which is*
* a graded
connected commutative algebra over IF.
26
Rings of Generalized and Stable Invariants
If a 2 A then there is a largest integer m(a) such that a 2 Im but a 62 Im+1 .*
* In this case
we say that a has filtration m(a). If a 2 A has filtration m and degree n then *
*we say a
represents the element a + Im+1 2 grI(A)m;n-m of total degree n. If a 2 A has*
* degree
n and filtration m and b 2 A has degree k and filtration j then
(a + Im+1 ) . (b + Ij+1) = ab + Im+j+1 2 grI(A) ;
so the product lies in grI(A)m+j; n+k-(m+j), and has total degree n + k.
PropositionP4.1ropositionP4.1roposition:4.1Let A be a graded commutative a*
*lgebra over a field IFand I A
a proper ideal. Suppose that I = (x1; : :;:xn). Then x1; : :;:xn is a regular s*
*equence in
A if and only if grI(A) = A=I x1; : :;:xn where x1; : :;:xn 2 I=I2 grI(A) den*
*ote the
residue classes of x1; : :;:xn.
ProofP:roofP:roofB:y induction on n. Consider the case n = 1, i.e., x = x1*
* 2 A is a nonzero
divisor of positive degree. The ideal Im is then the principal ideal (xm ) A *
*for m > 0.
Define ' : A -! Im by '(a) = a . xm . This map satisfies '(I) Im+1 and hence*
* induces
' : A=I -! Im =Im+1
which is an epimorphism. If a + I 2 ker', then by definition, axm 2 (xm+1 ), s*
*o axm =
bxm+1 for some b 2 A. Hence
0 = axm - bxm+1 = xm (a - bx) :
Since xm 2 A is not a zero divisor it follows a = bx so a + I = 0 2 A=I. Henc*
*e ' is a
monomorphism. Therefore the map
: (A=I) x -! grI(A)
defined by
((a + I) . xm) = axm ; m 2 IN
is an isomorphism, establishing the result for n = 1.
Let n 2 INand assume the result established for all ideals generated by a regul*
*ar sequence
of length less than n. Let I = (x1; : :;:xn) where x1; : :;:xn 2 A is a regular*
* sequence.
Set J = (x1; : :;:xn-1). By the induction hypothesis
grJ(A) = (A=J) x1; : :;:xn-1 :
The element xn + J 2 A=J is a nonzero divisor, so if K = (xn + J) grJ(A) is the
principal ideal generated by xn + J, then
grI(A) ~=grK(grJ(A)) ~=grK (A=J) x1; : :;:xn-1]~=(A=I) x1; : :;:xn
completing the induction step.
We turn next to the converse. Let x1; : :;:xn 2 A and (x1; : :;:xn) A the corr*
*esponding
ideal. Suppose
grI(A) = (A=I) x1; : :;:xn
27
Frank Neumann, Mara D. Neusel and Larry Smith
where x1; : :;:xn 2 I=I2 denote the residue classes of x1; : :;:xn 2 I. We must*
* show that
x1; : :;:xn 2 A is a regular sequence.
Consider the Koszul complex (see for example [11]x 5.2)
K = A E[u1; : :;:un]
where @(ui) = xi for i = 1; : :;:n. By Koszul's theorem ([11]theorem 6.2.3) we*
* need
to show that {K; @} is acyclic. To this end we filter K as follows: give A the*
* filtration
by the powers of the ideal I, give E[u1; : :;:un] the trivial filtration, and K*
* the product
filtration. In this way {K; @} becomes a filtered differential algebra. Let {Er*
*; dr} denote
the associated spectral sequence. Since the connectivity of Im goes to infinit*
*y with m the
spectral sequence converges to H*(K; @). From the definitions we have
E0 = grI(A) E[u1; : :;:un] = (A=I) x1; : :;:xn E[u1; : :;:un]
where
d0(ui) = xi fori = 1; : :;:n :
By Koszul's theorem ([11]theorem 6.2.3) we have {E0; d0} is acyclic, and
ae
(E1)p; *= H0(E0; d0) = A=I : p = 0
0 : otherwise.
Hence the term E1 is concentrated on the filtration zero line, so the spectral *
*sequence
collapses to yield the isomorphism
ae
Hi(K; @) = A=I : i = 0
0 : otherwise
___
as required. |__|
TheoremT4.2heoremT4.2heorem:4.2Let S GL (n; IF) be a set of pseudoreflec*
*tions and I(S) the
corresponding ideal of generalized invariants. Then
h i
grI(S)(IF[V ])= (IF[V ]=I(S))h1; : :;:hn
__gr
I(S)(IF[V ])= IF[h1; : :;:hn] :
___
ProofP:roofP:roofT:his is immediate from 2.9and 4.1. |__|
If S GL (n; IF) is a collection of pseudoreflections then IF[V ]=I(S) is a Poi*
*ncare duality
algebra (by e.g. [13]proposition 3 or [11]theorem 6.5.1). The algebra __grI(S)(*
*IF[V ]) depends
only on S and is called the ring of generalized invariants of S. By 4.2
__gr
I(S)(IF[V ]) = IF[h1; : :;:hn]
and although the polynomials h1; : :;:hn need not be well defined their degrees*
* are, and
we call deg(h1); : :;:deg(hn ) the generalized fundamental degrees of S.
Likewise for a representation % : G ! GL (n; IF) of a finite group G we call _*
*_grJ1 (G)(IF[V ])
the ring of stable invariants of G. From 3.4we obtain:
28
Rings of Generalized and Stable Invariants
TheoremT4.3heoremT4.3heorem:4.3Let % : G ! GL (n; IF) be a representation*
* of a finite group G.
If G is generated by pseudoreflections of order relatively prime to the charact*
*eristic of IF
then h i
grJ1 (G)(IF[V ])= (IF[V ]=J1 (G))h1; : :;:hn
__gr ___
J1 (G)(IF[V ])= IF[h1; : :;:hn] : |__|
ExampleE4.4xampleE4.4xample(4.4[11]x 8.2): Let IF be a field of characteris*
*tic p 6= 0 and for n 2 IN
consider the matrices2 3
1 0 . . . 0
660 1 0 . . .0 77
Ti= 666...... . . . 772 GL (n; IF) i = 1; : :;:n - 1 ;
4 0 . . .0 . .1. 0 75
0 . . .1 . .0. 1
where the off diagonal 1 is in the ith column of the last row. In other words,*
* if Ei;j
denotes the n x n matrix with a 1 in the ith row and jth column, and zeros else*
*where for
i; j 2 {1; : :;:n}, then Ti= I +En; ifor i = 1; : :;:n-1. These matrices are tr*
*ansvections
with the common direction En = (0; 0; : :;:1). Since the matrices T1; : :;:Tn-1*
* commute
with each other and all have order p, they afford a faithful representation of *
*E(n - 1), an
elementary abelian p-group of order pn-1.
Let i 2 IFx have order s, in other words i 2 IFis a primitive s-th root of unit*
*y, and let
H(i; r) GL (n; IF) denote the subgroup generated by T1; : :;:Tr and the matrix
2 3
1 0 . . .0
660 1 . . .07
64.. .. ..772 GL (n; IF) :
. . .5
0 0 . . .i
(This is the group denoted by G(i; r)* in [11]x 8.2.) The groups H(i; r) are*
* up to
conjugation a complete list of subgroups of GL (n; IF) satisfying dim(VH ) = n *
*- 1, where
VH = V=Span IF{hv - v | h 2 H; v 2 V }
is the module of covariants ([11]x 8.2) of the group H GL (n; IF). The invaria*
*nt theory
of the groups H(i; r) is computed in [11]proposition 8.2.13, where it is shown
IF[z1; : :;:zn]H(i; r)= IF[z1; : :;:zn-1; fs]
where Y
f = (zn + arzr + . .a.1z1) = cpr([zi])
2IFp
is the top Chern class ([15], [11]x 3.1 and x 3.2) of the orbit of zn. (In the*
* preceding
product IFp < IF is the prime subfield and z1; : :;:zn 2 V *is the dual of the *
*standard
basis of V = IFn.) Therefore
IF[z1; : :;:zn]H(i; r)= IF[zn]_spr:
(zn )
29
Frank Neumann, Mara D. Neusel and Larry Smith
The operation of the group H(i; r) on V *is given by
ae
Ti(zj)= zjz : j 6= n
n + zi: j = n
ae
Ti(zj)= zjiz : j 6= n
n: j = n.
Therefore the action on the coinvariants is given by
Ti(zn)= zn i = 1; : :;:r
Ti(zn)= izn
where zn 2 IF[V ]H(i; r)is the residue class of zn 2 IF[V ]. From this it follo*
*ws that
H(i; r) IF[zsn]
IF[V ]H(i; r) ~= ______spr
(zn )
and hence
IF[V ]H(i;r)2~=IF[zn]_(zs:
n)
Therefore ae
spr): i = 1
Ji(H(i; r)) = (z1;(:z:;:zn-1; zn s
1; : :;:zn-1;:zn)i = 2; 3; : :;:
and the ideal J1 (H(i; r)) coincides with J2(H(i; r)), i.e.,
J1 (H(i; r)) = (z1; : :;:zn-1; zsn) :
Let Di = TiTi 2 H(i; r) GL (n; IF) for i = 1; : :;:n - 1. Then direct computa*
*tion
shows that
2 3
1 0 . . . 0
6 0 1 0 . . . 0 7
(Di)k = 66. . 77= Tik+ k(i)En; i i = 1; : :;:n - 1
4 .. .. . . . 5
0 . . .k(i) . .0. ik
where
k(i) = i + i2 + . .+.ik :
Since i is a primitive s-th root of unity
0 = is - 1 = (i - 1)(! + i + i2 + . .+.is-1) :
Therefore s(i) = 0 and Di has order s for i = 1; : :;:n - 1. The matrix Di fix*
*es
the codimension one subspace IFn-1 IFn = V spanned by the first n - 1 standard
basis vectors. Hence Di is a pseudoreflection. Therefore H(i; r) is generate*
*d by the
pseudoreflections D = {D1; : :;:Dr; Ti}, all of order s, and s is relatively pr*
*ime to p,
which is the characteristic of IF. Hence by 3.4I(D) = J1 (H(i; r)) and
h i
grI(D)(IF[V ])= IF[zn]_(zsz1; : :;:zn-1; zsn = __grJ(H(i;(r))IF[V *
*])
n)
__gr s __
I(D)(IF[V ])= IF[z1; : :;:zn-1; zn] = grJ(H(i; r))(IF[V ])
30
Rings of Generalized and Stable Invariants
Note that
__gr ZZ=s __
I(D)(IF[V ]) = IF[z1; : :;:zn-1; zn] = grJ(H(i; r))(IF[V ])
where ZZ=s acts on IF[z1; : :;:zn] via the representation implemented by Ti.
31
Frank Neumann, Mara D. Neusel and Larry Smith
References
[1] J. F. Adams and C.W. Wilkerson, Finite H-spaces and Algebras over the Ste*
*enrod
Algebra, Annals of Math. 111 (1980), 95-143.
[2] J. L. Alperin, Local Representation Theory, Cambridge University Press, C*
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[4] M. Demazure, Invariants symetriques entiers des groupes de Weyl et torsio*
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* Reflec-
tions, in Progress in Mathematics 60: Geometry of Today, Roma 1984, Birkh*
*"auser
Verlag, Boston, 1985.
[7] F. Neumann, M. D. Neusel and L. Smith, Rings of Generalized and Stable In*
*variants
II, (to appear).
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*variants
III, (to appear).
[9] F. Neumann, M. D. Neusel and L. Smith, Rings of Generalized and Stable In*
*variants
IV, Preprint G"ottingen, 1994.
[10] G. C. Shephard and J. A. Todd, Finite Unitary Reflection Groups, Can. J.*
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6 (1954), 274-304.
[11] L.Smith, Polynomial Invariants of Finite Groups, A.K. Peters Ltd.. Welle*
*sley, MA
02181 USA, 1995.
[12] L. Smith, On the Invariant Theory of Finite Pseudo Reflection Groups, Arc*
*h. Math.
44 (1985), 225-228.
[13] L. Smith, A Note on the Realization of Complete Intersection Algebras by *
*the Coho-
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[14] L. Smith, P*-Invariant Ideals in Rings of Invariants, Forum. Math. (to a*
*ppear).
[15] L. Smith and R.E. Stong, On the Invariant Theory of Finite Groups: Orbit *
*Polyno-
mials and Splitting Principles, J. of Algebra 110 (1987), 134-157.
[16] L. Smith and R. M. Switzer, Polynomial Algebras over the Steenrod Algebra*
* : Vari-
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*) 27
(1984), 11-19.
[17] H. Toda, Cohomology mod 3 of the Classifying Space BF4 of the Exceptional*
* Group
F4, J. of Math. Kyoto Univ. 13 (1973), 97-115.
32
Rings of Generalized and Stable Invariants
[18] W.V. Vasconcelos, Ideals Generated by R-Sequences, J. of Algebra 6 (1967*
*), 309
-316.
[19] C.W. Wilkerson, Rings of Invariants and Inseparable Forms of Algebras ove*
*r the
Steenrod Algebra, (to appear in the Math. Proc. Camb. Phil. Soc.).
Frank Neumann, Larry Smith Mara D. Ne*
*usel
Mathematisches Institut Institut f"ur Algebra und*
* Geometrie
Bunsenstrasse 3 - 5 Postfac*
*h 4120
Universit"at G"ottingen Universit"at *
*Magdeburg
D 37073 G"ottingen D 39016 Magde*
*burg
neumann@cfgauss.uni-math.gwdg.de mara.neusel@mathematik.uni-magdebu*
*rg.de
larry@cfgauss.uni-math.gwdg.de
33
~~