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Annals of Mathematics, 105 (1977), 473-492

Matrices and cohomology

1 By LEONARD L. SCOTT*

In [15] Ree proved a theorem about permutations by making use of a

formula for the genus of a Riemann surface; an elementary proof sub-

i sequently was given in [5]. McKay [14] has shown that this theorem may

be used as a “Brauer trick”, that is, as a device for proving the existence

4

of conjectured proper subgroups of a given finite group.

In this paper we generalize Ree’s theorem to matrices. There are

several unexpected byproducts: some easily computed bounds for l-cohomo-

logy of finite or finitely-generated groups, a determination of the equivariant

cohomology of infinite polyhedral groups, a theory of polyhedral relation

modules including an explicit formula for their characters, a uniform treat-

ment of the Dickson invariant for an orthogonal group, and a Bockstein-

type theorem which relates 1 and Z-cohomology of irreducible modular re-

presentations of a finite group via decomposition numbers. Finally, we give

a general recipe for Brauer tricks, which shows that McKay’s Brauer trick

and the original version of Brauer [l] are special cases of a broader phe-

onomenon.

Some examples illustrating the application of these results have been

collected in the last section of this paper.

I would like to express my deep gratitude to Peter Hoefsmit, whose

expertise on Coxeter groups was invaluable in an early phase of this work,

I would also like to thank H. N. Ward for his collaboration on the Dickson

invariant, and Gordon Keller for his many pertinent suggestions which have

materially improved this paper.

Finally, I would like to mention that a generalization of Ree’s theorem

to Coxeter groups has been obtained independently by M. Herzog and G.

Lehrer.

1. The basic theorem

Let G be a group acting linearly on a finite-dimensional vector space V over an arbitrary field k. The group G may be finite or infinite, and we do

* Research partially supported by NSF.

474 LEONARD L.SCOTT

not insist that it act faithfully on V. For X a subgroup or element of G,

let v(X) = v(X, V) denote the codimension of the fixed-point space of X in

V. Also, we write v(X*) for v(X, V*), where V* is the dual of V.

THEOREM 1. Suppose G is generated by elements x,, l -a, x, with

xl- - - x, = 1. Then

Cz”=, V(G) 2 v(G) + v(G*) .

Proof. Let C be the k-space of n-tuples (v,, l . ., v,) with z)< e (1 - q)V. We have a linear map /3: V - C defined by

P(w) = ((1 - ⌧,)2), l l l , (1 - X?%>~) ,

and a linear map 6: C - V defined by

/

.

a(% * ’ -, w,) = ?I, + ⌧,wz + l l * + It�, * * - ⌧ - w + n 1 R.

Because of the identity

l- ⌧, . . . 2, = (1 - x,) + x,(1 - x,) + l *. + x, l l l x+1(1 - x,) ,

the image B of ,8 is contained in the kernel Z of 6.

The image of 6 is

l 3

(l-z,)V+x,(l-x,)V+ 0.. +x1*.*2,-1(1-3&)v,

which is clearly just (1 - x,)V+ . . . + (1 - x,) V, the smallest kG-submodule

of V with trivial action on the quotient. Thus the image of 6, and hence

C/Z, has dimension v(G*).

On the other hand, the kernel of ,8 is obviously the space of fixed-points

for G in V, and so the image B has dimension v(G).

Finally, (1 - xJV clearly has dimension v(x,), and so the dimension of

C is CL, v(xJ. Thus

Cb, v(x,) = dim C = dim B + dim Z/B + dim C/Z

2 dim B + dim C/Z

= w(G) + w(G*) . Q.E.D.

Remarks. a) Reefs theorem is obtained immediately by taking G to

be a group of permutation matrices: v(X) is just the degree minus the

number of orbits of X.

b) One can in some sense take into account eigenvalues other than 1 by

multiplying the xi’s by a system of scalars ei satisfying E, . l l E, = 1. The

group G is just replaced by a subgroup of G x k”.

In the same spirit, one can get further inequalities, for G and zl, . . ., x,

fixed, by tensoring V with any finite-dimensional kG-module.

’ The map 6, applied to all of V(m), is sometimes called the “Fox derivative.”

MATRICES AND COHOMOLOGY 475

c) Obviously both sides of the inequality in Theorem 1 are additive with

respect to direct sums of kG-modules. So any time McKay’s “Brauer trick”

works for a permutation representation, it will have to work for one of its

ordinary irreducible constituents (which will then be a constituent of the

permutation representation of the larger group on the cosets of the conjec-

tured subgroup).

Moreover, Theorem 1 can be used as a “Brauer trick” in characteristic

p: the left-hand side is computable in terms of the Brauer character if

$1, l l l t X, are all p’-elements. (Indeed, Thm. 1 gives a necessary condition

that a given class function be the Brauer character of an irreducible module.)

At worst, the left-hand side can be computed from the Jordan canonical

form of x,, ..a, x,.

d) The proof of Theorem 1 generalizes easily to actions on abelian

groups, not just finite-dimensional vector spaces, and a corresponding result

may be formulated. In fact the argument even applies to actions on non-

abelian groups, so long as G acts trivially on V/Z(V); I have not pursued

this. A generalization of Theorem 1 in a different direction is considered in

Section 6.

Schur indices. Suppose k = C, G is finite, and 2 is the character of V. Let h(X) denote the nonnegative integer which is the difference between the

two sides of the inequality in Theorem 1. Let p” denote the generalized

character xb, (p - pi) - 2(p - l), where ,o is the character of the regular

representation and pi is the character of G induced from the trivial character

of (x,). Application of Frobenius reciprocity yields immediately that

h(x) = (p”, xl .

Since h(X) 2 0, p” is in fact a character of G. Moreover, p” is afforded by a

rational representation since this is true for p and each of the pi (cf. [16],

Prop. 33). Of course, the Schur index ma(x) of any irreducible character x

must divide its multiplicity in any rationally-afforded character. This proves

the following corollary:

COROLLARY. SupposeGisajinitegroupand x isanordinaryirreduc-

ible character of G. Then m,(x) 1 h(X).’

In Section 3 we shall give an explicit realization of p” as the character

of a ZG-module, namely, the relation module for the polyhedral group as-

t As a condition on Schur indices, this corollary gives no more information than m&)l(~, ,oc)=(x, l)(q and ?n&)Ix(l). (I am grateful to I. M. Isaacs for correcting my blindness on this point in an earlier version of this paper.) The conditions in S6, however, appear more subtle.

476 LEONARD L.SCOTT

sociated with x,, ..., x,.

In the next section we are concerned with conditions in arbitrary char-

acteristic on the difference between the two sides of the inequality in

Theorem 1. One further result is obtained which applies when G is finite

and k = C, namely that h(x) is always even when x is real-valued.

2. The homology invariant

We return to the notation of Theorem 1, and in addition let h(V)=

h(G, V) = h(G, x,, . . . , x,, V) denote the nonnegative integer which is the

difference between the two sides of the inequality in Theorem 1. We call

h(V) the homology invariant of V with respect to G and x1, . . a, x,.

Also, we define @(G, V) to be the subgroup of H’(G, V) consisting of

classes which restrict to 0 in each H1((xi), V). Thus @(G, V) = H’(G, V) provided H’( (x,), V) = 0 f or each i-for example, if each xi has finite order

not divisible by the characteristic of k.

PROPOSITION 1. a) We have h(V) 2 dim I?‘(G, V), and h(V) 2 dim I?‘(G, V*).

b) Moreover, if H”(G, k) = 0 and each xi has jinite order not divisible by the characteristic of k, then

h(V) 2 dim H’(G, V) + dim H’(G, V*) .

Proof. Let M be a k-space of l-cocycles which maps isomorphically onto

@(G, V). Define a kG-module structure on 7 = M @ V by x(7 @ v)=

r $ v + r(x) for x e G. It is easily checked that the fixed points for G in v

are contained in V, and that the extension v is split when restricted to any

of the (xi>. Thus the right-hand side of the inequality in Theorem 1 grows

by at least dim I?‘(G, V), while the left-hand side remains the same. That

is, h(v) 5 h(V) - dim I?‘(G, V). So h(V) 2 dim E?‘(G, V). By symmetry

we also have h(V) = h(V*) 2 dim E?‘(G, V*). (Note that v(x) = v(x*) when

x in an element of G.) This proves part a). We remark that a proof could

also be given by direct examination of the quotient Z/B which appears in

the proof of Theorem 1 (cf. the proof of Prop. 3 in S 3).

Now consider part b). Continuing with the notation of the previous

paragraph, we obtain information on H’(G, p) from the long exact sequence

of cohomology:

H’(G, 8*) - H’(G, V*) - H”(G, (p/V)*) .

Since G acts trivially on p/V, and H’(G, k) = 0 by hypothesis, we have

H’(G, (v/V)*) = 0. Thus

h(v) = h( v*) 2 dim H’(G, v*) 2 dim H’(G, V*) ,


which, combined with the inequality h(V) - dim F(G, V) 2 h( 8) obtained

above, gives the desired result.

Remarks. a) Obviously, this proposition can be used as a bound on

1-cohomology. Conversely, it improves the effectiveness of the Brauer trick

when the relevant 1-cohomology groups are already known for the larger

group. See [ll] for a list of known 1-cohomology groups when G is a finite

group of Lie type.

b) The assumption that H”(G, k) = 0 in part b) of Proposition 1 is not

entirely necessary, and it is possible to say more precisely what is required:

Suppose we have a central extension E: 0 - k -% G - G - 1. Then for each

z( there is a unique element 55, in G having the same order as IU$ and mapping

onto xi (since the order of zi is not divisible by the characteristic of k).

Thus we have an element .&?Z) 6 k defined by

f(E) = 2, l * * Ft?, .

It is easily checked that f” is k-linear and takes the same value on equivalent

extensions. Hence f” induces a map

fi H”(G, k) - k .

Now suppose f = 0. Then we claim that the conclusion of part b) holds. Let P be the polyhedral group (m,, . . . , m,) where m, is the order of xi.

Thus P is generated by symbols x,, .. ., x, subject only to the relations

$1 * . . . x, = 1 and z;i = 1. If we apply the above paragraph to P, we obtain a map H”(P, k) -f k, which is clearly injective.’ Moreover, the

map f: W(G, k) -k factors (in general) as H”(G, Jc) - H”(P, k) -+ k, where

the first map is inflation; hence in the present case the inflation map H”(G, k)-

H”(P, k) is 0.

Now a diagram chase shows, in the notation of the proof of part b),

that each element in the (isomorphic) image of H’(G, V*) in H’(P, V*) can

be pulled back to H’(P, v*). Hence h( 8*) = h(P, v*) 1 dim H’(P, P*) 2 dim H’(G, V*), and the proof of part b) can be completed as before.

PROPOSITION 2. Suppose G preserves a quadratic form on V whose associated bilinear form is nondegenerate. Then h(V) is an even integer.

Proof. In fact we will show that the quotient Z/B in the proof of

Theorem 1 carries a natural nondegenerate symplectic form under these

hypotheses. Let & be the given quadratic form and b its associated bilinear form.

Since each x, preserves b, the form b induces a nondegenerate pairing

’ We will show in $3 that this map is in fact an isomorphism when P is infinite.

478 LEONARD L. SCOTT

b,: C; x C, -+ Ic, where C: denotes the quotient of V by the space of fixed points of xi, and C, denotes the space (1 - x,)V. Hence we have a non-

degenerate pairing C b,: C” x C --*kwhereC* =CCrandC=zC,.

Let ,6? and 6 be the maps defined in the proof of Theorem 1. These maps induce maps ,B*: C* - V and a*: V ---t C* defined by

p*[vJ = (1 - x;‘)z)~ for [v,] e CT, and 6*v = Cr=, [XL?:-‘, - - - x;‘v] for v E V .

The image of 6” is, of course, precisely the annihilator of 2 with respect to

the pairing C bi, and the kernel of ,B* is precisely the annihilator of B. Our

next aim is to exhibit an isomorphism d of C* with C which carries 6* into p, and -p* into 6. This will at least give us a nondegenerate form on Z/B.

Define d = (dij) where dij: CT - Ci is given by

dJw] = (1 - X&I

I

0 ifi<j,

ifi= j,

(1 - xJ(1 - x;‘v) if i > j .

The map d is clearly an isomorphism. Moreover, the ith component of d8”v is

(1 - x&2, * * * x;’ 1) + ci>j (1 - X$)(1 - x;‘)x;Il * * * x;%

= (1 - xJx;I1 * * * x;lw + (1 - x,)(1 - X,r_ * * * x;l)v

= (1 - !I?& ,

and so da* = p. To see that 6d = -p*, we evaluate 6d on an element [v] 6 12’;:

6d[w] = z, - - - q-1(1 - %>V + c*>i Xl - * - x,Jl - x,)(1 - x;‘)v

= -x, * ’ * Xj(1 - x;‘)v + (x, * *. Xj - l)(l - x;‘)w

= -(l - x;‘)?J= -@*[VI .

We now have a nondegenerate form on Z/B, which is most convenient-

ly described on Ker P* by

(u, w) = (C b,)(u, dw) = xi b(U<, (1 - X&u,) + cl>j b(%, (1 - X,)(1 - X?‘)Wj)

= c, b((1 - Xil)Ui, wl) + ci>j b((1 - Xil)Ui, (1 - x;‘)wJ ,

where u = C [ui], w = C [wi]. At last, the quadratic form comes into

play: if u = w, we have

0 = Q(Ci (1 - x;l)ui) = Ci Q((l - Xil)UJ + Ct>j b((1 - Xi’)Ui, (1 - x;‘)uJ .

Also, we have the identity

Q(v) = &WV) = Q((x-‘-1)~ + V)

= Q((x-l-1)~) + Q(V) + b((x-‘-l)v, V)

= Q((l - X-‘)v) + Q(V) - b((1 - II;-+, V)

for ‘u e V and x preserving Q; thus Q((l - 6)~) = b((1 - X-‘)v, v). Hence


(u, u) = 0. Q.E.D.

When the xi each have finite order not divisible by the characteristic of

k, then the existence of a nondegenerate symplectic form on Z/B may be

established without calculations using the cup-product H’(P, V*) x H’(P, V)+ H2(P, k) E k, where P is the polyhedral group of the previous remark. This

does not seem to work in general.

My original proof of Proposition 2 used unpublished work of H.N. Ward

on the Dickson invariant. Ward proved, using the geometry of orthogonal

groups, that the map which sends an element x in the orthogonal group of

Q to the residue class (mod 2) of V(X) is always a homomorphism, and that V(X) is congruent (mod 2) to the number of terms in any expression for x

as a product of reflections. (Such an expression always exists if k is large

enough. The term “reflection” here means a map v - v - b(r, v)/&(r), where

r e V satisfies Q(r) f 0.) These results are essentially equivalent to Pro-

position 2, since v(G) + v(G*) is even (=2v(G)) when V is self-dual as a

kG-module. The present proof of Proposition 2-essentially, the existence of the

map d-was found experimentally, after it was determined that the exist-

ence of d was highly desirable for the cohomology and homology theory of

polyhedral groups developed in Section 3. It would be nice to have a natural

homological characterization of d, even if a more natural construction is

not possible. For example, one might speculate that d induces the cap-

product isomorphisms H,(H, Z) z H’(H, Z) for all surface groups H of finite

index in P, and is in some sense unique with this property. See [21] for

related material.

COROLLARY. Suppose G is jinite, and x is a real-valued character of G. Then h(X) is even.

Proof. Since h(X) = h(X*) we may assume that x is irreducible. If

m,(x)= 2 the result follows from the corollary in Section 1. But if m,(x)= 1,

then x is afforded by an orthogonal representation, and Proposition 2

applies. Q.E.D.

It should be noted that neither Proposition 2 nor its corollary can have

any influence on the effectiveness of the Brauer trick, if the v(x,) are all

completely known, since these results apply as well to whatever subgroup

of the larger group that the xi happen to generate, and the right-hand side

of the inequality of Theorem 1 is automatically even in both cases.

3. Cohomology of polyhedral groups

We keep the notation of the preceding sections, and again assume we

480 LEONARDL.SCOTT

are in the situation of Theorem 1. In addition, we let F denote the free

group on symbols z2, ..., x, (which may also be viewed as the group on

symbols x,, . . . , x, subject to the single relation x, . . . x, = 1. Also, we let

P denote the polyhedral group (m,, . . a, m,) where mi is the order of xi in

G (possibly mi = -). That is, P is the group on symbols x,, . . . , x, subject

to the relations x, . . . x,=land~~i=lifm~<c~.

PROPOSITION 3. We have h(V) = dim @(F, V) = dim I?‘(P, V).

Proof. Let g& Z’(F, V) be the k-space of cocycles which represent

elements of I?‘(F, V). If 7 e g, then r-0 on (x,), so 7(x,) e (1 - xJV. More-

over, the cocycle condition gives CE, $I* . . x,-,7(x,) = r(x, . * s x,) = r(1) = 0.

Thus we have a map

from g to the space 2 defined in the proof of Theorem 1. This map is clearly

injective, and we claim that it is an isomorphism: If (v~, . . ., v,) e 2, define

an action of F on ,% @V by xi(a @ V) = a @ xiz, + a~, for each i (including

i =.l). This action is well-defined because Cy=, x, . . . x~-,z)~ = 0. Hence the

equation x(1 @ 0) = 1 @ r(x) defines a 1-cocycle 7 on F with ?‘(xJ = vi; more-

over, r-0 on (xi) since vui e (1 - xJV. Thus we have an isomorphism .??s 2.

Clearly the coboundaries correspond to B under this isomorphism, and

so @(F, V) z Z/B. In particular dim E?‘(F, V) = h(V). When each xi has finite order mi, the same argument applies for P; it

is only necessary to observe that %?(a @ V) = a@ v since ZI$ e (1 -xi) V. Q.E.D.

The proof of Proposition 1 of course holds when V is any abelian group;

that is, I?‘@‘, V) = fi’(P, V) = Z/B if the groups 2 and B are constructed

as in the proof of Theorem 1. We shall now use this fact to construct a

simple Z-resolution for P, if P is infinite. (The first three terms are valid

for P finite as well.)

Let a,: ~~=, ZP - ZP be the sum of right multiplications by the elements 1 - xi of ZP. The image of 3, is obviously the kernel of the augmentation

6: ZP+Z. Let E be the set of indices i for which mi < 00, and for i 6 E let

Na: ZP 4 ZP be right multiplication by the sum of all members of (2,).

Now define

by &:ZP@C,ZP- CL, zp

a,(~ 43 CafE vi) = (C;z, 21x1 - a. xi-l) + CjEE v,N .

The image of d, is obviously contained in the kernel of d, (again by the

identity CL, x, * . . x,-,(1 - 2,) = 1 - s1 . a. x~). We claim, of course, that


equality holds:+

Suppose not, and let p be a prime with k @ (Ker d,/Im a,)#0 for k= Z/pZ.

Since the image of d, is Z-free, we may regard k @ Ker a, as a subspace of

Ker 10 a,, and so Ker 10 &/Im 10 3, f 0. Next, define 2, and 3, as the

maps induced by 10 3, and 10 & in the diagram

lc30 32 I c (1-G) /I

kP -p- c;=, kP(l - x,)a, kP .

Since the kernel of C (1 - xi) is precisely the image under 10 d, of c, kP,

we must have Ker &/Im 3, f 0. Finally, take k-duals of the bottom row, and

identify (Cy’, kP(1 - xi))* with Cy=, (1 - x,)kP*. As may be easily verified,

we now have precisely the sequence that defines Z/B for kP* (with 6 = 2,X

and p = 2:). Thus I?‘(P, kP*) r (Ker &/Im a,)* # 0. But this is absurd,

since H’(P, kP*) S H,(P, kP)* = 0. Thus the image of 3, is indeed the kernel of a,. The kernel of 8, has two

possible forms, depending on whether P is infinite or not: Clearly

C, ZP(l - x,) is contained in the kernel of a,, and is precisely the intersec-

tion Ker 3, n C, ZP. Suppose now we have an element v @ C, vi E Ker a,.

If some xj has infinite order, then j 4 E, and so vx, . . . xjW1 = 0, which forces

v = 0. If each Xj has finite order, then we have vx, . * * xidl = VjNj for each j zx 1, . . . . n. This gives v = vx, = vxz= . ..=vx. and so v is fixedbyP. If

P is infinite this forces v = 0, and if P is finite, then v must belong to the

diagonal copy of Z in ZP (and indeed any such v can arise). In the infinite case the remaining terms of the resolution are obvious:

take&tobethesumof right-multiplicationsC,(l-xi): C, ZP-ZP@c, ZP (with zero projection onto the first ZP), and take a, to be the sum of right-

multiplications C, Ni: C, ZP - C, ZP, etc. To summarize:

PROPOSITION 4. Suppose P is infinite. Then we have a resolution* of Z by free ZP-modules:

. ..-~.zP~C,ZP~ZP~~,ZP~~~zPa,zP~z

where the maps & are as described above. Also, the first three terms (in-

volving E, a,, and a,) are a partial resolution when P is jinite.

It is an easy matter to convert Proposition 4 into a description of

+ Stammbach and Gruenberg point out that a more natural proof of exactness can be given by viewing Ker & as a relation module, cf. 8 4. The construction of the first three terms of a resolution for a group is standard, and apparently due, essentially, to Lyndon [12].

* This resolution has appeared in the literature before; cf. [19].

482 LEONARD L.SCOTT

cohomology:

THEOREM 2. Suppose P is infinite, and let A be any abelian group on

which P acts. a) For r 2 3 we have H’(P, A) z CT=, Hr((xi), A) by restriction, and

Cyzl H,((xJ, A) rr H,(P, A) by corestriction (i.e., “+nclusion”).

b) The restriction map H’(P, A) -+ CL, H’((xJ, A) is surjective, and

an isomorphism if any xi has infinite order. In general, its kernel is na-

turally isomorphic to the cokernel of the natural map

C;z, H’(Cd A) = Ci eE H&xi), A) 83 CiEE Iz,(<xJ, A) - Ho@‘, 4 .

Dually, the corestriction map Cy=, Hz((x,), A) - H,(P, A) is injective, and an isomorphism if any x, has infinite order. In general its cokernel

is naturally isomorphic to the kernel of the natural map

H”(P, A) - C,,. H’((xih A) CT3 CiEE ~“((xJ, A) g CL, H&i), A) .

c) The image of the restriction map H’(P, A) - EYE, H’((xJ, A) is the kernel of the map Et, H’((x~), A) - H,(P, A) described above, and its kernel is naturally isomorphic to the group I?‘(P, A) = Z/B described in the proof of Proposition 3 and Theorem 1.

Dually, the kernel of the corestriction map CL, H,((xJ, A) - H,(P, A) is the image of the map H”(P, A) -xb, H,((xJ, A) described above, and its cokernel is also naturally isomorphic to @(P, A).

The proof is fairly straightforward from Proposition 4, except for the

very last sentence, and so I will omit the other details. This is how @(P, A) appears the second time in part c): If one works out directly the cokernel

of the map Cy=, H,((xi), A) - H,(P, A) from Proposition 4, the result is

homology of a complex

A - C A/HO((xJ, A) - A

where the maps are a -Cl=, [x;?, l . . x;la] and Cb, [a,] -Et, (1 - x;‘)a+

If we apply the isomorphism d described in the proof of Proposition 2, this

complex becomes a complex defining I?(P, A), with even the appropriate sign

change according to a standard convention for dualizing complexes. Q.E.D.

COROLLARY. Suppose P is injinite, and A is an abelian group on which P acts. Assume in addition that each m, is jinite, and that multiplication by mi on A is invertible. Then

a) We have H,(P, A) = H’(P, A) = 0 for r 2 3. b) There are natural isomorphisms H,(P, A) E H”(P, A) and H”(P, A)E

H,(P, A). (In particular, if char. k does not divide any of the m,, then

H”(P, k) E k.)


c) The groups H,(P, A) and H’(P, A) are both naturally isomorphic to

the group E?‘(P, A) previously described.

The jinite case. In the finite case we can extend the partial resolution

a,, a,, E by one more term a;, defining 3;: ZP @ Cb, ZP - ZP @ j-& ZP to

be d, = C 1 - oi on the second factor, and N@ CL, - T, on the first,

where N is (right-multiplication by) the sum of all the elements of P, and

Ti is a sum of left-coset representatives for (xi) in G.

Passing to cohomology we obtain a natural exact sequence

0 - I?‘(P, A) - H’(P, A) = C;z, H1((xO, A)

z C;=, fio((xi), A) 5 Iz,(P, A) - H”(P, A)

= C;=, fP((x:i), A) .

PROPOSITION 5. If P is Jinite, and acts on an abelian group A, then we have for each integer r an exact sequence of Tate cohomology groups

I?‘(P, A) = CE, f+((xA A) - ii-“(P, A) - &+l(P, A)

= C;=, ~T+l((x,>, A) ,

where the second map is corestriction if we identify l?((x,), A)= &-2((xi), A). The map &+(P, A) -+ Iz’+l(P, A) may be viewed as cup- product with a generator for H3(P, Z); the latter group is cyclic of order 1 P~/l.c.m.{mi}, which is either 1 or 2.

Proof. The first assertion follows immediately from the exact sequence

above by dimension-shifting.

The kernel of H3(P, Z) % C H3((xi), Z) = 0 is thus isomorphic to the

cokernel of c @((x,>, Z) z B”(P, Z), which is just

Z/c (1 PI/m,)Z = Z/g.c.d. (1 Pj/m,)Z = Z/(1 Pi/l.c.m.{mc})Z .

If G is a generator for H3(P, Z), then, of course, c = B U 1 where

1 e @P, Z). Now let Y be a direct sum of copies of Z, considered as trivial

ZP-modules, which has the group of fixed-points in A as a quotient. The

map @(P, Y) + @(P, Y) is clearly cup-product with ~7, and Y -, A induces

a surjection Izo(P, Y) - I+(P, A), so B”(P, A) -+ p(P, A) must also be cup-

product with o. Now it follows by dimension-shifting that each map -7 2 H-(P,A)-H ^‘+l(P, A) is cup-product with C.

The finite polyhedral groups are (ignoring obvious reorderings of the

m,, and any terms mi = 1) just the groups

(m,m) = Z/m& (2,2, m) = D,,, (2,3,3) = %,,(2,3,4) = O,, and (2,3,5) = &,

484 LEONARDL.SCOTT

cf. [3] and [4]. The corresponding group orders are m, 2m, 12, 24, and 60. We immediately compute from the above that the order of G is 1, (2, m), 2,2,

and 2, respectively. In particular Q has order 1 or 2, and the proof is com-

plete. Proposition 5, together with the fact that the kernel of the restriction

map H’(P, A) -+ C H1((xt), A) has the same description @(P, A) as in the

infinite case, gives a fairly complete description for H1 and Hz, and gives

H3 up to the kernel of the transfer map C H’((xJ, A) - H’(P, A). In

particular we note the following:

COROLLARY. If P is finite, and acts on an abelian group A for which I?(A) = &(A) = 0 (which is certainly true if A has no trivial ZP-submodule or quotient module # 0) then:

a) The restriction map H’(P, A) - Cb, H’((xi), A) is surjective, and its kernel is the group @(P, A) previously described.

b) The restriction map H’(P, A) -CT=, H’((x,), A) is an isomorphism. c) The restriction map H3(P, A) -XL, H3((xJ, A) is injective, and

its cokernel may be viewed as the kernel of the corestriction map

C;=, H’((xJ, A) - H’(P, A). 7% e image of the latter is just m.H’(P, A),

where m = 1 Pl/l.c.m.{mi}.

The last assertion follows from the surjectivity in part a), and the fact

that restriction to a subgroup followed by corestriction gives multiplication

by the index.

Similarly one could describe H’(P, A) under the same hypothesis: H”(P, A) has a subgroup isomorphic to H’(P, A)/m.H’(P, A), with quotient isomorphic

to the kernel of the corestriction map xb, H’((x$), A) - H”(P, A), which

has image m.H’(P, A). If, in addition, A = V is a finite-dimensional vector space, then such

arguments suffice to determine dim Iz’(P, V) for all integers r. I leave the

details to the interested reader. It is interesting to note that a) and c) above

force h(V) = 0 when char. kl;m (and A”(P, V) = &,,(P, V) = 0). At the same time, we note that h(x) = 0 for any ordinary character of

a finite polyhedral group P, since H’(P, V) = 0 for the associated module.

The genus. In Ree’s proof of his theorem in the case of a transitive

permutation group, the difference between the two sides of the inequality

in Theorem 1 turns out to be 2g, where g is the genus of a certain Riemann

surface. It is worth pointing out that this interpretation can be completely

recovered from Proposition 3, though I will not give full details:

Suppose V is the permutation module associated with a transitive action


of G on a finite set a. Regard the group F described at the beginning of

this section as acting on Q, and let H be an isotropy subgroup. Using Burn-

side’s procedure [2, pp. 384-51 we may construct a properly discontinuous

topological action of F (and hence of H) on R” such that the quotient

X = R’/H is a surface which is obtained from a compact surface X by stick-

ing a finite number of pinholes in it. As a kF-module, V is induced from

the trivial l-dimensional module k for H, and so H’(F, V) E H’(H, k). On

the other hand, since R” is contractible and H acts properly, we have the

well-known result H’(H, k) z H’(X, k). Thus H’(F, V) g H’(X, k). When

H’(F, V) is modified to I?‘@‘, V) in this isomorphism, it turns out that H’(X, k)

must be modified to H’(X, k). Thus dim HI@;, k) = h(V) is independent of k

(cf. § 1, Remark a)). In particular X isorientable, and 2g = dim H’(X, k) = h(V)

where g is the genus of X. Q.E.D.

4. The relation module

We keep the notation of the preceding section, and in addition let

M, = RJR; where R, is the kernel of the natural surjection P--t G. Also,

we set M, = RJR> where R, is the kernel of F c, G. Both M, and M, are

naturally endowed with the structure of ZG-modules, and the structure of

M, has been extensively studied by Gruenberg [7], [8], and Gaschiitz [6], at

least when G is finite. We shall give a description of M, in terms of M,, in

the sense that we shall describe the kernel of the natural surjection M,-+M,. Finally, we will show, when G is finite, that C @ Mp affords the character

p” described in Section 1, and we conclude this section with a structure

theorem for S @ M,, where S is a p-adic ring.

I am indebted to David Wigner for conversations which motivated the

following lemma; cf. also [17, § l] and [18, p. 2101.

LEMMA 1. Let N be a normal subgroup of a group Y, and set F = Y/N. a) If Z E is regarded as a right ZY-module (and left Z F-module), then

we have an isomorphism N/N’ z H,( Y, Z P) of left Z F-modules.

b) If {y&e A is a set of generators for Y, then the ZF-module N/N’ is

canonically isomorphic to the cokernel of the natural map KY - Kf;, where

K,is the kernel of the sum of right-multiplications CA (l- yJ: C, Z Y-Z Y,

and K, is dejined analogously.

Proof. Since Hr( Y, ZP) = 0 for r 2 1, the homology spectral sequence

gives an isomorphism H,( Y, Z F) g H,( y, H,(N, Z P)). We have H,(N, Z Y) g Z Y Qz N/N’, and so

H,( 7, H,(N, Z 7) ‘: Z P &y N/N’ z N/N’ .

486 LEONARD L.SCOTT

This proves a). To prove b), we recall that H,( Y, Z Y) is, by definition, iso-

morphic to the kernel of Z Y Qy C, Z Y -+ Z Y&Z Y modulo the image of

Z Y&KY - Z Y Ozy C, Z Y. That is, HI( Y, Z Y) is isomorphic to the cokernel of ZF@&Ky - K?. Since KY - KF factors by a surjection through this map, we have part b).

The description of M, in the theorem below is known, cf. [17, § 11.

THEOREM 3. The ZG-module M, is isomorphic to the kernel of the sum of right-multiplications zb, 1 - xi: CL, ZG - ZG, and, if we regard M, C Cy=, ZG, then the kernel of the surjection M, - M, is precisely the set of elements CL, (a, - a,x, . . . xi-J where ai belongs to the set Ai offixed- points for x, in its right-action on ZG (i = 1, . . ., n).

The kernel of MF - M, is thus isomorphic as a ZG-module to the direct sum CE, Ai when G is in.finite, and to (EYE, A&Z when G is finite.

Moreover, M, and n/r, are both free as Z-modules, and in particular

MT - Mp is Z-split.

Proof. The isomorphism of M, with the kernel of cy=, 1 - xi follows

from part b) of Lemma 1 and the well-known fact that the elements 1 - xi

form a ZF-basis for the augmentation ideal of ZF. (Thus the kernel of

Cy=, ZF --+ ZF is 0.)

Again by part b) the kernel of M, -+ M, is the image of the kernel of

CE, ZP- ZP in CL, ZG. However, we know the kernel of Cy=, ZP- ZP by Proposition 4: it is the set of elements Ci fE v,N, + CL, vx, . . . xipl where

v and the vt belong to ZP. (Recall that E is the set of indices i for which

m, -Cm, and Ni is the sum of all elements of (xi) when i e E.) This gives

the desired description for the kernel of M, -M,, if we note that A, = (ZG)N,

forieE,whileAi=OforieE.

Of course, M, is Z-free by the Nielsen-Schreier theorem, and the same

is true for M, by analogous results for Fuchsian groups (cf. [9] and [lo]).

This completes the proof of Theorem 3.

COROLLARY. If G is jinite, then C @ M, affords the character p” described in Section 1.

Proof. From Theorem 3 we see that C @ MF affords the character

(n - 2)~ + 1, while the kernel of C @ M, - C @ Mp affords the character

(Et, pi) - 1. H ence C @ M, affords

(n - 2)~ + 1 - (Cr=, ,Q() + 1 = Cy=, (p - pi) - 2(p - 1) = P” . Q.E.D.

Now keep the assumption that G is finite, and let S be the ring of local

integers in a p-adic number field with residue field k = S/KS = S. Write


S @ MF = I @ Q, and S @ M, = 1p @ Qp, where Q,, Q, are projective

SG-modules, and I, I, have no projective direct summand # 0. Gaschiitz

and Gruenberg have shown that I does not depend on the presentation F-G; cf. [7]. We shall show that this is also the case with Ip in favorable circum-

stances:

THEOREM 4. Assume G is jinite, that p$rn$ for each i, and that

H’(G, k) = 0. Then Ip is isomorphic to I@ I*, where I* = Hom,(l, S).

For the proof, we first require the following lemma:

LEMMA. Under the hypothesis of Theorem 4, if 0 -A -E - G -+ 1 is

an estension of G by a jinitely-generated SG-module A, then the canonical map P c--f G can be lifted to a map P-E.

To prove this, it is enough to show that the inflation map H”(G, A) -

H’(P, A) is 0. If we regard H,(P, A) as a trivial SP-module, then the map

A- H,(P, A) induces an isomophism H”(P, A) E H’(P, H,(P, A)) by the

corollary to Theorem 2. Hence we may assume to start that A is a trivial

SG-module, and even that A = S or S/n”S since A is finitely-generated.

However, H”(G, Sjn’S) = 0, by an obvious induction and the fact that

H”(G, k) = 0; also H”(G, S) 4 H”(G, S/VA) is injective for sufficiently large

A, since G is finite, so H”(G, S) = 0 as well. So indeed H”(G, A) - H’(P, A) is 0. This completes the proof of the lemma. We will now prove the theorem.

Proof of Theorem 4. The natural map M,-M, induces a map I---&@ M,.

We also have a map S @ Mp --*I, which comes from lifting the map P-G to

the canonical extension E of G by I (possible by the lemma). We claim that

the composite 0: I - I is an automorphism: Clearly $ is the restriction of a

map E -* E which induces the identity on the quotient G; thus E/Im 4 is a

split extension of G by I/Im $. Since (E, I) is essential [7], we must have

Im $ = I, and so # is an automorphism.

Consequently, I - S @ M, is a split injection. Let T be the inverse

image in S @ M, of a complement T, to the image of I in S @ M,. Thus

S @ M, = I + T, and In T = 0. In particular, T g Q is projective. Also,

T contains the kernel of S @ MF- S@ M,, namely, S @ N where

N r EYE, A,jZ is the kernel of MF - 2M, described in Theorem 3. Observe

that S @ Ai is projective, since j (xi) 1 = mi is invertible in S; thus

S @ NM Q, @ Q,/S, where Q, is projective, and Q, is the unique projective

indecomposable SG-module containing a trivial submodule f 0.

The inclusion S @ N = T is S-split, since N & M, is Z-split by Theorem

3. Passing to duals and taking a projection, we have a homomorphism

T* - (Q,/S)* with kernel Q: @ T& Since (C: ZG)/MF is isomorphic to the


augmentation ideal, we also have a surjection C,” SG - (Q,/S)* with kernel

isomorphic to the direct sum of S @ n/r, = 1@ Q with a projective module.

Now by Schanuel’s lemma and the Krull-Schmidt theorem, we have

T; M IT@ Q, where Q, is projective. Taking duals once more, we have

SOMP M I @ T, M I @ I* @ Q,“, which proves the theorem.

5. A Bockstein theorem

We shall describe some higher-dimensional analogues of the module I

which was just discussed, and give homological formulas for their characters.

The nonnegativity of multiplicities then yields a general system of ine-

qualities, relating the dimensions for cohomology groups of irreducible

modular representations via decomposition numbers. The same system

could also be obtained without the character formulas by Bockstein-type

methods; hence the title of this section.t

Our notation and assumptions are as follows: G is a finite group and p is

a fixed prime. K is a p-adic number field, S is the ring of local integers in

K, and k = SjnS = S is the residue class field. We shall assume for con-

venience that K is sufficiently large, so that all irreducible KG or kG-modules are absolutely irreducible. For each irreducible Brauer character $ of G,

we let L, be an irreducible kG-module affording #. If x is an irreducible

ordinary irreducible character of G, we write as usual x = c, d,,$ on

$-elements.

Let . . . -Q, 2 Q, 2 Q, 5 S be a resolution of S by projective SG-modules

which is “minimal” in the sense that (QO, E) is a projective cover of S, (Q,, a,)

is a projective cover of the kernel of E, etc. Such resolutions have the fol-

lowing property, which I learned from Jon Alperin:

(5.1) If V is an irreducible kG-module, then Hom,(Q,, V) z Hi(G, V) .

The point is that ai (and E) must map the Frattini quotient of Q$ iso-

morphically onto the Frattini quotient of the kernel of the next term, and

so Ker ai = Im d,,, is contained in the Frattini submodule of Q,. A homo-

morphism 7: Qi - V thus automatically satisfies the cocycle condition

c?$+~Y’ = 0, and does not factor through 8% unless 7 = 0.

We can rephrase (5.1) as follows: Let QO, <D, ... be the characters

affordedbyK@Q,, K@Q,... . Then if $ is an irreducible Brauer character,

we have

(3.2) (CD%, $) = dim H*(G, L,) .

t J. A. Green suggests that perhaps the term “Euler characteristic” might be more appropriate.

MATRICESANDCOHOMOLOGY 489

The point here is that Hom,(Q,, L,) has dimension equal to the multi-

plicity of L, in the Frattini quotient of Q,. This in turn is the multiplicity of the indecomposable projective module corresponding to L, as a direct

summand of Qi, and the well-known character-theoretic expression for this

multiplicity is (Of, $).

Now, define Ii as the kernel of ai, and let pi be the character afforded

by K @ Ii. (The module 1, is isomorphic to the module I discussed in the

previous section; cf. [7].) W e may also define I0 and ,6?, in the obvious way.

THEOREM 5. For each irreducible character x f 1 and each integer

r 2 0, we have

. Also,

(-1)’ C, dxr Cl=, (- l)‘dim HYG, -&I = (P,, x) 2 0 .

(- 1)’ CL, (- l)idim H”(G, k) = (,63,, 1) 2 0 .

The inequality for r = 2 is worth being stated by itself:

COROLLARY. If x f 1 is an irreducible character, then

C, d,,(dim H’(G, L,) - dim H’(G, L,) + dim H”(G, L,)) 2 0 .

Also, dim H”(G, k) 2 dim H’(G, k).

COROLLARY. If V is an irreducible kG-module which can be lifted (to an S-free SG-module), then dim H”(G, V) 2 dim H’(G, V).

Proof of Theorem 5. Expressing ,6, in terms of the @‘i and 1, we have

(-l)?P, + 1 = c;=, (-‘)“a( .

Taking inner products and applying (5.2) yield Theorem 5.

6. A recipe for Brauer tricks

In this section we briefly consider a generalization of Theorem 1. In

particular, we are able to put McKay’s Brauer trick and the original Brauer

version under one theoretical roof.

Suppose G is a group generated by subgroups X1, . . ., X,. Again, sup-

pose G acts linearly on a finite-dimensional k-vector space V, and define v(X)

as in Section 1.

Let C be the direct sum xi B’(X,, V) of 1-coboundary groups, and let

,6?: V + C be the map which takes v e V to the sum of the coboundaries

a,,~: R;* + v - xiv, xi E Xi. The reader will observe that, so far, our notation

essentially agrees with that in the proof of Theorem 1.

Now let r be a relation in G among elements of the various Xi, taken

in any order, with repetitions allowed. Thus, r has the form r1 l . . rrc = 1,


where rj e Xstj, for some assignment s of indices. Then we have a well-defined

map L: C - V taking

Cb, &,w, to c,“=, r1 * - - ?.jel(?“j - l)V,,j, .

Just as in the proof of Theorem 1, the image of ,L? is contained in the kernel

of L. Hence, if R is a collection of such pairs (r, s), and we set v(R)=

dim (Wl~r,s~eR Ker a,,,), then we have

(6.1) C,,, 4-K) 2 v(G) + WI .

This generalizes Theorem 1 in a sense, though of course, the difficulty

lies in estimating v(R). (However, the calculation of w(R) is just a linear

algebra problem in a specific, known representation, and can probably be

put on a computer.)

The original Brauer trick can be phrased as the simple assertion

GG) + VW,) 2 v(G) + 4% n X,)

if G = (X1, X2). Each element y of X, n X2 gives a relation 7616-l = 1 with y

regarded as in X,and y-l in X2. If R contains these relations, then ncr,,, Ker a,,,

is easily seen to be contained in the kernel of the natural map C--B’(X,, V)+ B’(X, n X2, V), and so the above inequality is a consequence of (6.1).

It is also worth pointing out that several of the results of the preceding

sections go through in this more general setting. In particular, if one defines

h,(V) to be the nonnegative difference between the two sides of (6.1), then

h,&V) may be interpreted as the dimension of I?‘(P, V) where P is the free

product of the X, with the relations R factored out (and as before @(P, V) is the kernel of H’(P, V) -C, H’(X,, V)). If G is finite and x is an ordinary

irreducible character of G, then h,(x) is defined as fjR( V) for V a CG-module

affording x. And again, h,(x) is the multiplicity of x in the character of

C @ M, where M is the relation module associated with the presentation

P- G; the proof is by the the Hochschild-Serre sequence. (In particular,

ma(x) I Mx).) Th is relation module, of course, may not be Z-free; indeed,

suitable choices of R will generally yield for P a number of finite extensions

of G.

Finally, I mention that the entire procedure could be modified by

replacing B’(X,, V) in the definition of C by any subgroup of Z1(Xi, V) con-

taining B’(X,, V). Of course, v(X,) is replaced in (6.1) by the dimension of

this subgroup. If one takes always Z1(Xi, V) itself, then @ gets replaced

by H’ in the previous paragraph. In particular, if R is large enough, then

h,(V) becomes precisely dim H’(G, V). A procedure similar to this has ac-


tually been used by Martineau [13] to calculate some specific l-cohomology

groups. See also Lyndon [12].

7. Examples

We give a few easy sample applications of some of our results. First,

an application of Theorem 1:

PROPOSITION. The group SL(a, 3) is not a (2, 3, 7) group for n = 6 or 9.

Proof. Let us take n = 9 first. Let X, y, x e G, where G = SL(9, 3),

have orders 2, 3, 7 with xyx = 1, and suppose G = (x, y, x). Let W be the

standard g-dimensional module for G, and set V = W @ W. Thus G has no

fixed points on V or V*, and so v(G) = V(G*) = 81. The minimum possible

dimensions of fixed point spaces for x, y, x on V are 4’ + 5’ = 41, (l/3)81 = 27,

32 + 6 = 15. Hence h(v) 5 81 - (41 + 27 + 15)= -2, a contradiction.

The case n = 6 is similar: h(v) 5 36 - (20 + 12 + 6)= -2, again a con-

tradiction. This proves the proposition.

Next, we illustrate Proposition 1 of Section 2 as a bound on 1-cohomology

to prove a special case of a result of Graham Higman.

For a statement and proof of the general result see [20].

PROPOSITION. ExtB( W, W) = 0 where S= SL(2, 2”) and W is the standard

2-dimensional module.

Proof. Clearly we can take n>l. Let G be the usual upper triangular

Bore1 subgroup of S. Then it suffices to show H’(G, V) = 0 where

V = Hom(W, W). We have G = (x, y), where x= (f i-l), Y = ({ &), and

a generates GF(2”)“. Choose ,& # a-‘, p # 1, so that x, y, and x = y-lx-l

are all 2’ elements. Then

h(V) = v(x) + v(y) + v(x) - v(G) - v(G*)

=2+2+2-3-3=0.

Hence H’(G, V) = 0. Q.E.D.

Finally we give an illustration of (the first corollary to) Theorem 5 as a non-vanishing criterion for 2-cohomology. The following result has been

obtained independently by Jon Alperin.

PROPOSITION. Let V be the absolutely irreducible module in charac-

teristic 2 for SL(2, q), q odd, that is associated with the quadratic residue code (dim V = (q - 1)/Z). Then H’(G, V) # 0.

Proof. We recall that the structure of the permutation module W=lZ

of dimension q + 1 is as follows in char. 2: W has unique maximal and


minimal submodules M, L and W/M w L w l,, M/L M V @ V” where 0 is a

group automorphism. In particular H’(G,V) w H’(G, V”) f 0. In char. 0

the character of l”, is 1 + x, x irreducible. Hence the irreducible Brauer

characters (char. 2) in x are those affording l,, V, V”. Of course H”(G, 1,) = 0,

so Theorem 5 gives

r

dim H”(G, V) - dim H’(G, V)

+dim H”(G, V”) - dim H’(G, V”)

I

20.

+ dim H”(G, lG)

That is, B(dim H”(G, V) - d im H’(G, V)) 2 - 1, and so H”(G, V) # 0. Q.E.D.

UNIVERSITY OF VIRGINIA, CHARLOTTESVILLE

REFERENCES

[ 1 ] R. BRAUER, Representations of finite groups, Lectures on Modem Muthematics, Vol. I, edited by T. Saaty, John Wiley and Sons, 1963.

[ 21 W. BURNSIDE, Theory of Groups of Finite Order, second edition, Cambridge University Press, Cambridge, 1911.

[ 31 H. S. M. COXETER AND W.O. MOSER, Generators and Relations for Discrete Groups, Springer-Verlag, 1965.

[ 41 P.M. CURRAN, Cohomology of finitely presented groups, Pacific J. Math. 42 (19721, 615- 620.

[ 51 W. FEIT, R. C. LYNDON, AND L. L. SCOTT, A remark about permutations, J. Combina- torial Theory 18 (1975), 234-235.

[ 61 W. GASCH~TZ, Uber modulare Darstellungen endlicher Gruppen die von freien Gruppen induziert werden, Math. Z. 60 (19541, 274-286.

[ 71 K. W. GRUENBERG, CohomoZogical Topics in. Group Theory, Springer Lecture Notes 143 (1970).

181 -, Uber die Relationmoduln einer endlichen Gruppe, Math. Z. 118 (1970), 30-33. [ 91 A. H. M. HOARE, A. KARASS, AND D. SOLITAR, Subgroups of finite index of Fuchsian

groups, Math. Z. 120 (1971), 289-298.

I101 p, Subgroups of infinite index in Fuchsian groups, Math. Z. 125 (1972), 59-69. [ll] W. JONES AND B. PARSHALL, On the 1-cohomology of finite groups of Lie type, Proc.

(Park City) Conf. Finite Groups, 1976, 313-327. [12] R. C. LYNDON, Cohomology theory of groups with a single defining relation, Ann. of

Math. 52 (1950), 650-665. [13] R. P. MARTINEAU, Splitting of group representations, Pacific J. Math. 45 (1973), 571-576. [14] J. MCKAY, to appear. [15] R. REE, A theorem on permutations, J. Comb. Theory 10 (1971), 174-175. [16] J.-P. SERRE, Representations Lineaires des Groupes Finis, 2nd edition, Hermann, 1971. 1171 R. G. SWAN, Groups of cohomological dimension one, J. Alg. 12 (1969), 585-610. [18] B. ECKMANN AND U. STAYMBACH, On exact sequences in the homology of groups and

algebras, Ill. J. of Math. 14 (1970), 205-215. [19] S. MAJUYDAR, A free resolution for a class of groups, J. London Math. Sot. (2) 2 (1970),

615-619. [20] E. CLINE, B. PARSHALL, L. SCOTT, Cohomology of finite groups of Lie type I, Publ. Math.

I.H.E.S. no 45, 169-191. [21] R. BIERI, Gruppen mit PoincarC-Dualit%, Comment. Math. Helv. 47 (1972), 373-396.

(Received April 7, 1976)

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