§23 Mod P Reflection Groups

166

§23: MOD P REFLECfION GROUPS

As of yet no one has obtained a classification for W reflection groups p

analogous to those obtained in §21 and §22 for k provided one assumes that

Icl

= m,~

and

~

However,

is prime to p then Clark and Ewing obtained

strong restrictions by relating W reflection groups to the already disp

cussed p-adic reflection groups. In fact,

the Clark-Ewing classification

for p-adic reflection groups simply carries over to W reflection groups. p In this chapter we outline their results. §23-1:Main Results Clark and Ewing treat W reflection groups by reducing to the results p

for

~

reflection groups described in §22.

THEOREM (Clark-Ewing) Let C be a finite group of order prime to p. Then C has a representation as a W reflection group if and only if C has p a representation as a

~

reflection group.

The correspondence is established by using Zp representations. Let C have order prime to p. One shows that, given a representation p: C there exists a representation p: G duces a

~

~

~

GL (W ), n

p

GL (Z ) inducing p. In turn, p inn

p

representation. As we will see,

reflection representations to

~

this construction sends W

p

reflection representations. Moreover we

obtain, in this manner, a one to one map between W reflection represenp tations and

~

reflection representations.

This correspondence between reflection groups carries over to their invariant theory. This is not surprising. For, since

Ici

is prime to p,

C c CL(V) being a reflection group is equivalent to its ring of invariants S[V]G being polynomial. We will show PROPOSITION: Let C be a finite group of order prime to p. Given a repre-

167

VI: Reflection Groups and Classifying Spaces

sentation p: G

GL (~ ) then the following are equivalent: n p G (1) ~p[tl.···,tn] = ~p[xI"",xn] where Ixil di •

~

A

A

= Wp[x1 ....• Xn]

where Ixil

~[tl'···.tn]G = ~[Xl.···.Xn]

where Ixil

(ii) Wp[t l,·· .,tn]G (iii)

d

i

= di pl. It follows

(The action of G in (ii) and (iii) is that induced from

from the proposition that we can determine the polynomial ring of invariants which arise from W reflection groups of order prime to p. The answer p

is the same as for \

reflection groups. The table of degrees given in

§22-2 for irreducible p-adic reflection groups applies to irreducible W p reflection groups as well. One must only add the extra restriction that the degrees {d ITd i

=

... ,d are prime to p. This restriction is forced because l, n} IGI t 0 mod p.

To summarize.

the W reflection groups of order prime to p have. as p

invariants. polynomial algebras W [x •..•• x ] of type {d ••..• d } where: p

(i) ITd

1

n

1

n

is prime to p

i

(ii) {d •..• d

is an union of the sets of degrees appearing in the last n} 1 column of the table in §22-2

All the above only applies when IGI is prime to p. There is very little known about the modular case Le. when p divides IGI. Certainly. the

~

reflection groups do not give the answer. There are W reflection groups p which are not p-adic reflection groups. The example GL (W ) is discussed n

p

in §23-4. §23-2: Representation Theory

As already stated. the link between W and \ p through

~p'

is obtained by passing

The following two lemmas provide a 1-1 correspondence (provid-

ed the group has order prime to p). In all that follows we will always work with free -

~

p

modules.

LEMMA A: Suppose G has order prime to p. Then (i) any representation p: G

~

GL (W ) lifts to a representation p: G n

p

168

TheHomowgvofHopfSpac~ ~

(ii) if P1,P2:G ~ GLn(Wp) have liftings P1,P2:G ~ GLn(Zp) and PI ~

P2 then PI

=P2· ~

Proof:(i) Given P we want to find P so that GL (Z )

.i-:

G

n

<.

commutes. Since Z

p

= ~p lim Z/ r

1

GL (W )

P

Ps : G

p

p

n

we have GL (Z ) np

GLn (Z/p r) we want to find Ps+ 1: G

~

P

~

=< lim GL (Z/ r). --np

So, given

GLn (Z/ p r+1) so that

1 GL (Z/ r+1)

~n

G

p

1

0s G Ln (Z/p r) We have a commutative diagram 1

1

K

K

1

1

1

G'

1

1

c

GL (Z/ r+l) n p

c

GL (Z/ r) n p

"Y

ps(G)

1

1

1

where

1

1

K

= Ker

...

G' = ...- l(p (G» s

Now K is a abelian p group and ps(G) has order prime to p. Thus 2(P H

= 0 and the short exact sequence 1 ~ K ~ G' ~ ps(G) ~ 1 splits s(G);K) as a semi-direct product G' = ps(G) ~ K. Consequently, we can define Ps + 1" ~

(ii) Think of PI and P2 as Wp[G] modules V and W and PI and P2 as Zp[G] "

A

A.

"

modules V and W. So V and Ware the mod p reductions of V and W. Given a W [G] module isomorphism 0: V p

isomorphism isomorphism

n:

V ~ W. If

=W we can certainly

~

lift it to a Z

p

module

we average 0 over G we obtain a Z [G] module p

169

VI: Reflection Groups and Classifying Spaces

covering O. Q.E.D. We also have a relation between

representations and lp representa-

~

tions. Before establishing the relation let us observe some consequences of Lemma A. It tells us that direct sum splittings of f

Zp [G] modules.

respond to direct sum splittings of

p

[G] modules cor-

Since (IGI.p)

=I

it is

a standard fact that every fp[G] module decomposes into irreducible modules (i.e. f

vector spaces V such that only {O} and V are invariant under

p

G). It follows that l p [G] modules likewise decompose where an irreducible is a free l

p

module such that the only G invariant submodules are {O} and

pSM for s ~ O. LEMMA B: Suppose that G has order prime to p. (i) Any representation representation (ii) If

° 1,°2 :

A

0:

0:

G

G

~ GLn(~)

is conjugate to a

A

A

GL (l ) C GL (ID )

~

n

n

p

'p

° 1,°2 :

G ~ GLn(~) are conjugate to A

~

G ~ GLn{lp) and

A

~

01 ~ 02 as ~[G] modules then 01 ~ 02 as lp[G] modules.

Proof: (i) Pick any "lattice" L

= (Zp )n

C

(~)n. Now consider

= Ell

L'

gL gEG

We claim that L'

is an invariant lattice. Obviously, it is invariant. We

must show that L'

= (lp )n . A

1 = l p [-] P

A.A

Since ID

P

i t follows that any lp module

is of the form F Ell T where F is free and T consists of p torsion. The inclusion L' C (~)n tells us that L' is a free lp module. Since L'0 ~ =' A nAn

we must have L'

(~)

sentation

= (lp)

. The invariant lattice L' defines the repre-

0.

( i i) We can reduce to the case of irreducible l [G] modules. We have G p

A

A

invariant lattices M and N such that M 0 ~ _ N 0 ~ as ~[G] modules. Then M C M 0

~ ~

N0

~

is a lattice of N 0

A

~.

S

So p MeN for some s

~

O. But every non trivial l [G] submodule of N is of the form ptN. So M ~ N p

as l [G] modules. Q.E.D. p

§23-3: Proof of Theorem 23-1 and Proposition 23-1

170

The Homoloqv of Hopf Spaces

lei

Since

is prime to p, G being a reflection group is equivalent to

its ring of invariants being polynomial. Hence, Theorem 23-1 follows from Proposition 23-1. The proposition is easy to prove. Proof of Proposition: (i

il

(it il

<=>

Obviously. (ii) ator Av(x) image.

==>

= ~ L gx gEG

Consequently.

(iii). Regarding (iii)

==>

(ii) the averaging oper-

is a projection operator with the invariants as its the

ring

a

direct

summand of

~

The averaging operator can be used to show that 7L [t ..... t J p l n

G

-->

§23-4: The Dickson Invariants As we mentioned in §23-1 groups and

~

the correspondence between IF p

reflection groups does not hold when p divides the order of

the group. GL (IF ) provides an example of a group which is a IF n

p

group but. in most cases. has no representation as a The IF

reflection

~

p

reflection

reflection group.

reflection group property is seen from the fact that the invariants

p of GL (IF ) form a polynomial algebra. n

p

Dn where Iq. I 1

= pn

= IFp[tl •...• tnJ

GL (IF ) n

p

= IFp[ql •.... ~_lJ

- pi-I. This algebra is called the Dickson invariants It

has played a major role in algebraic topology in recent years. See Wilkerson [13J for a convenient summary of the properties and uses of the Dickson invariants