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Chapter 6: Bilinear Forms on G-Algebras
Chapter 6
Bilinear Forms on G- Algebras We have now come t o the last leg of our long journey into the theory of Galgebras. This chapter contains some basic information concerning bilinear forms on G-algebras. Here we concentrate exclusively on the case where the ground coefficient ring is a field of prime characteristic p. The bilinear forms that we consider below are far from being nonsingular. The case where the ground coefficient ring is a complete discrete valuation ring requires a different technique which can be found in a work of Thdvenaz (1988a). In the first section, we develop some general properties of bilinear forms on G-algebras with an eye to future applications t o group algebras. It turns out that the most convenient setting t o work with is that of G-algebras endowed with G-stable bilinear forms. These forms are somewhat analogous t o that introduced by Green (1983). The main results are contained in Sec.2. In Sec.3 we concentrate exclusively on the group algebra case and introduce a bilinear form whose rank is equal t o the number of blocks of F G with defect group P. Here F is a field of characteristic p > 0 and P is a given p-subgroup of G. T h e chapter also contains some additional results which are of independent interest. To facilitate the reading, important definitions have been repeated where necessary.
Bilinear Forms on G-Algebras
1
Preliminary results
In what follows, F denotes a field and all vector spaces over F are assumed t o be finite-dimensional. From Lemma 1.2 onwards, it will be assumed that charF = p > 0. For convenience of exposition, the section is divided into three subsections.
A. General facts Our aim here is to provide some general information concerning bilinear forms on G-algebras. We also introduced a specific form that will be studied in the next section. Let V be a vector space over F and let
be a symmetric bilinear form. Then the rank of f , written r a n k ( f ) is defined by rank( f ) = d i r n F ( v / v L ) where
v L ={x E V l f ( x , V ) =
0)
It is clear that the rank of f is equal to the rank of the matrix
where a;j = f(vi,vj) and vl,. . . ,v, is a basis of V . Our point of departure is the following elementary lemma concerning the rank of symmetric bilinear forms.
Lemma 1.1. Let f : V x V -+ F and let f; : V, x V, -+ F be symmetric bilinear forms, 1 i i n . Suppose that for each i E (1,. . . ,n ) , there is an F-linear map n; : V + V, and a nonzero pi E F such that the following two properties hold : (i) The induced map
<
1 Preliminary results
is surjective.
(ii) f (x, Y)= Cy=l~ i f i ( n i ( x )ri(Y)) , Then we have n
for all X ,Y E V
Proof. We may harmlessly assume that each pi = 1. Consider the symmetric bilinear form
defined by
n
Then, owing t o (ii), we have
It therefore follows from (i) that, for any given x E V,
Hence r induces an F-isomorphism V/VL
as desired.
--+
(ny=&)/(nE1
K)'. Thus
.
From now on, we fix a finite group G and a finite-dimensional G-algebra A over a field F of characteristic p > 0. We also assume that A is endowed with an F-linear map X:A-+F which is symmetric in the sense that X(ab) = X(ba) for all a , b E A and also G-stable in the sense that
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If A is an interior G-algebra, then as X we can always choose the character of a matrix representation of A over F. In what follows, P denotes a p-subgroup of G and Br$ : AP
--,
A(P)
is the corresponding Brauer morphism. We remind the reader that, by Lemma 1.7.1, both AP and A(P) are NG(P)/P-algebras and Br$ is a homomorphism of NG(P)/P-algebras over F. The following observation will allow us to introduce a bilinear form associated with P, G, A and A.
Lemma 1.2. The following properties hold : (i) lier(Br;) C I i e r A. (ii) X induces an F-linear map Xp : A(P) -t F given by Xp(~r;(a)) = X(a)
for all a E AP
(iii) Xp is symmetric and Nc(P)/P-stable.
Proof. It is clear that (ii) and (iii) follow from (i). To prove (i), we first note that, by definition
where, by convention, for P = 1 we have l i e r ( ~ r ; ) = 0. Assume that & is a proper subgroup of P and let a E A;. We must show that X(a) = 0. Write a = ~ r ; ( x ) for some s E AQ and let T be a left transversal for Q in P. Then, since X is G-stable, we have
as required. Following Brouk and Robinson (1986), we now define the bilinear form
""
~ P , G: A$
x A$
--,
F
as follows : if a, b E A$ with b = ~ r g ( b ' )for some b' E A ~ then ,
1 Preliminary results
357
Our next aim is t o record some elementary properties of the introduced bilinear form. AX Lemma 1.3. The form fp;G is well defined, symmetric, and associa-
tive. Proof. Assume that a = ~ r g ( a ' )and b = Tr?(b1) for some a', b' E A'. Let the F-linear map Xp : A(P) -+ F be defined as in Lemma 1.2(ii). Then
Observe also that, by Lemma 1.7.2(i),
Thus we must also have Br$(a) = Tr,N"'P'lP
(B$ (a'))
and Br;(b)
T$ = Tr,N ~ ( P ) / P ( ~ (bl))
(2)
Replacing G, P, A by N G ( P ) / P , 1, A(P), we may therefore assume that P = 1. Because X is G-stable, we have X(g(al)b) = /\(a' 9-'b) for all g E G. Thus
Consequently, if Trf(bl) = Tr?(bt1), then X(abl) = X(abl1) and so the form is well defined. Since X(abf) = X(bfa), the form is symmetric by virtue of (3). Moreover, because cb = ~ r f ( c b I )for all c E AG, we have f;lih(ac, b) = X(acbl) = f s ( a , cb) This shows that the form is also associative, as required. B Owing t o Lemmas 1.2 and 1.3, we may now define the bilinear form
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with the role of P, G, A and X played by 1, N G ( P ) / P , A ( P ) and X p , respectively. The following observation explains how the new bilinear form can be brought into argument. For all a , b E A:, we have
Corollary 1.4.
Proof. Write b = ~ r $ ? ( b ' )for some b' E A ~ Then . A(P),XP
f,,,(p,,p(~rf
(a), ~ r f ( b ) ) = ~ p ( ~ r S ( a ) ~ r S ( b ' ) )(by (2)) = Xp(~r$(abl)) = X(abl) (by Lemma 1.2(ii))
= f;$(a,b) as required. W Next we record the following elementary property. Lemma 1.5.
For any integer n
> 1, the ordinary trace map
is the only (up to scalar multiples) symmetric F-linear map from M n ( F ) to the field F. Proof. Let V be the F-space of all symmetric F-linear maps M,(F)
+
F . We must show that dimFV = 1. But, for any q E V ,
and hence the result.
.
dim^ Mn ( F )/ [ M n ( F ) Mn (F)] = 1,
For any point a of A, let X , denote the character of A afforded by the corresponding simple A-module Ae/J(A)e where e E a. Of course, X , is independent of the choice of e . We let X(a) denote X(e) (which depends
1 Preliminary results
359
only on a , not on the particular e chosen). In what follows, pt(A) denotes the set of all points of A. Lemma 1.6. Assume that F is a splitting field for A and that J ( A )
Ker A. Then
C
A=
c
X(a)xa
aEpt(A)
, A(a) be the unique block of A/J(A) Proof. Given a E P ~ ( A ) let which contains the point a J ( A ) of A/ J(A). Then, by Proposition 3.1.2,
+
is a block decomposition of A/J(A). On the other hand, since F is a splitting field for A, it follows from Corollary 11.1.4 in Vol.1 that, for any given a E pt(A), A(a) Mn, (F)
>
for some integer n, 1. Because J ( A ) 2 Ker X and, by Lemma 1.5, there is (up to scalar multiples) only one symmetric F-linear map Mn,(F) -+ F, we may write
It is easy t o see that x,(,f3) = Sap for all a , p E pt(A), Thus pa = X(a) for each a E pt(A), as required. W
B. The group algebra case To break the monotony, we now offer an important example pertaining t o the G-algebra F G where G acts on F G by conjugation. We remind the reader that F is a field of characteristic p > 0 and P is a p-subgroup of G. For any conjugacy class C of G, S(C) denotes a defect group of C (with respect t o p). For any subset S of G, S+ E F G denotes the sum of all elements in S. If HI, Hz are subgroups of G, then we write H1 GG H2 to indicate that H1 is G-conjugate to a subgroup of Hz. Of course, HI CG H2 means that H1 is G-conjugate to a proper subgroup of H2. Let C1,. . . ,Cn be all conjugacy classes of G with S(C;) CG P, where P is a given p-subgroup of G. Then, by Corollary 1.6.5,
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360
form a basis of ( F G ) ~Keeping . the above notation, we now record
Example 1.7. Let X : FG + F be defined by X ( ~ , ~ ~ = x X~I , g ) x, E F . Then (i) X is an F-linear map which is symmetric and G-stable. Moreover, for any subsets X , Y of G ,
(ii) For any X , Y E {C1,.. . ,C,), we have FG,X
f p , ~( X t
,Y + )=
I X l p ~ / l G l p ~ if
y = x-' and 16(X)I = /PI otherwise
(iii) ((FG):)' = l i e r ( ~ r Fn~( )F G ) = ~ V where V is the F-linear span of all C t , where C is a conjvgacy class of G with 6 ( C ) C G P. Proof. (i) It is clear that X is an F-linear map which is symmetric and G-stable. By definition of A , we also have
where X(xy) = 1 if x = y-'
and X(xy) = 0 if x f y-'.
Thus
as required. (ii) Choose g E Y and a Sylow psubgroup Q of CG(g)such t h a t Q G P. Let C be the P-conjugacy class of g. Then
TT;(C') = ( C G ( S ): C p ( g ) ) Y t and (CG(g): C P ( g ) )is a p'-number since Q C Cp(g). Hence
and therefore
1 Preliminary results
361
# X-',
then X n C-*= 0 and so f pFGG' X (x+ , Y + ) = 0. Hence we may assume that Y = X-', in which case X n C-' = C-l. If 16(X)I < [ P I , then FG X p divides IC-ll = ( P : Cp(g)). Hence, in this case, fP,G' ( X f ,Y t ) = 0. Finally, assume that 16(X)I = [PI. Then Q = P and so
If Y
which forces
as required. (iii) Given a nonzero z in ( F G ) ~ , write x = X I Xt~ - - . XrX: for some nonzero A; E F and some XiE {C*,.. . ,C,), 1 5 i 5 r . Then, by if and only if IS(X;)I < [PI for all 1 5 i 5 r (which (ii), x E ((FG)?)' means that x E V). The latter is equivalent to the requirement that each P-conjugacy class in X i , 1 5 i 5 r , is nontrivial, i.e. t o x E K e r ( ~ r ; ~ ) (see Lemma 1.12.9).
+
Keeping the notation of the preceding result, we now record
Proposition 1.8. Let e be any central idempotent of F G . Then the rank of the bilinear form
is given by
Proof. P u t V = (FG)?~. Since (FG): it is clear that V (FG)?. We claim that
is an ideal of ( F G ) ~= Z ( F G ) ,
Since
V nl i e r ( ~ r ; ~ )
c
(FG)$ n l i e r ( ~ r F ~ )
= ((FG>~>'
(by Example 1.7(iii))
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362
it is clear that V fl ~ i e r ( ~ r 2 F v' ~. ) Conversely, let xe E V' for some z E (FG):.
Then, for any y E (FG)?,
since, by Lemma 1.3, f[gt"s associative. Thus, by Example 1.7(iii), r e E ~ e r ( B r ; ~ which ) proves (4). We now deduce that
.
where the last equality is a consequence of Lemma 1.7.2(ii) and our identification of ( F G ) ( P ) with F C G ( P ) (see Corollary 1.7.13). We close this subsection by recording the following consequence of the above result.
Corollary 1.9.
The bilinear forms
have the same rank.
Proof. Setting H = NG(P), we obviously have N G ( P ) = N H ( P ) and C G ( P ) = CH(P). Applying Proposition 1.8 twice (with e = l ) , we have
as desired. H
1 Preliminary results
363
C. Comparison with a bilinear form of Green
Again, we assume that A is a finite-dimensional G-algebra over a field F of characteristic p > 0. We also assume that A is endowed with an F-linear map X:A+F which is symmetric and G-stable. In addition, we assume that (A, A) is a symmetric algebra, i.e. we require that K e r X contains no nonzero left (equivalently, right) ideal. As before, P denotes a p-subgroup of G. Our aim is t o show that the rank of the bilinear form
introduced prior to Lemma 1.3 is equal to the rank of a bilinear form introduced in a work of Green (1983) (see Corollary 1.12 with H = P). All FG-modules below are assumed to be finitely generated. Given an ~)) FG-module V and a subgroup H of G, we put V$ = ~ r g ( ~ n v ( Vwhere
is the relative trace map. Recall that Inv(VH) denotes the subspace of all H-invariant elements of V , while Inv(V) is the subspace of all G-invariant elements of V. As usual V* = HomF(V, F) is the contragredient to V. Thus V* is an FG-module via (g f)(v) = f(g-'v) for all v E V, g E G, f E V*. For the purpose of this subsection, we need only to prove that
for all v E Inv(VH), cr E Inv((V*)H). We shall derive this property as a consequence of a more general result that will be needed later. Let U and V be FG-modules. Recall that HomF(U, V ) is an FG-module via (gO)(u) = g(O(g-lu))
for all
u
E U,g E G,O E H o r n ~ ( UV) ,
Then HomFH(U, V) is the space of all H-invariant elements of HomF(U, V), HomFG(U, V) is the space of all G-invariant elements of HomF(U, V) and
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364
Of course, ~ r Z ( ~ o r Vn )~) is~a (subspace ~ , of H o m F G ( U , V ) . Let l H be the trivial FH-module, i.e. l H = F and H acts trivially on F, and let M = ( 1 ~ ) Then ~ . M is a transitive permutation FG-module which has an F-basis { u , ~ x E G I H ) ( G / H is the set of all left cosets of H in G) on which G acts by
gu, = ugX
for all g E G, x E G/H
The basic element u = U H is H-invariant and M = F G . u. In what follows, F = 1~ is the trivial FG-module. Consider the maps
t E H o ~ F G ( MI G, ) , 6 E H o ~ F H ( M1 , ~
)
defined as follows :
( c ,cx E F ) . We now fix an FG-module V and prove the following result. Lemma 1.10. Furhter to the notion above, put E = E n d F G ( M ) . Then (i) There is a homomorphism p : E -+ F of F-algebras such that
where 8 E E and 8 ( u ) = ~ x E G I HcxuX. Moreover, 87 = p(O)q, ( 8 = p(O)( for any 0 E E . (ii) The map HonzFG(V,d4)-+ H o m F H ( V ,l G ) ,f H 6f is an F-isomorphisr (iii) The nzap HonzFc(M,V) + H o m F H ( l G , V ) ,g H gE is an F isomorphism. (iv) Given f E HomFG(V,M ) , g E H o m ~ ~ ( ?vd ), ,put a = 6 f, a' = ge, P = t f and p' = gq. Then (a) a P ' ( 1 ~=) P a ' ( 1 ~ ) . (b) p = T r $ ( a ) ,/3' = T r E ( o 1 ) .
1 Preliminary results
365
Proof. (i) It is clear that the spaces HOmFG(lG,M), HomFG(M, 1 ~ ) are, respectively, left and right E-modules (acting by composition of maps). Moreover, these spaces are one-dimensional; in fact HomFG(lG,M ) = F . q and HomFG(M, l G ) = F . (. Therefore there are homomorphisms p , p' : E + F of F-algebras such that 0q = p(Q)q, t 0 = pl(0)<
for all 0 E E
(1)
We claim that p = p'. Indeed, we first check that 6q(lF) = 1~ = l ~ ( 1 ~ ) . Then from (1) we obtain
Finally, a direct calculation shows that SOq(lF) = <0&(lF) = cz, where O(u) = C c,u, (observe that an element 0 E E is determined by 0(u), since M = F G . u). This implies (i). (ii) and (iii) This is a special case of Proposition 18.1.6 in VoI.1. (iv) From (1) (putting B = f g ) , we obtain a string of equalities : p(fg)
= 6f977(1~)= aD'(lF)
= t f g ~ ( 1= ~ )D a l ( l ~ ) proving (a). It is a consequence of the definitions of 7 , E, [ and 6 that TT$(S) = <, T T ~ ( E=) q. Hence
and Dl
= gq = ~ T GT ~ (=ET) TG ~ ( ~=ETT$(CY') )
proving (b). W As an application, we now record
Corollary 1.11. Let V be an FG-module and let H be a subgroup of G. Then (i) POTany a E H o ~ F H ( V l, ~ ) a', E H o m ~ ~ ( lV), c,
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(ii) For any v E Inv(VH), a E Inv((V8)H), we have
Proof. (i) By Lemma l.lO(ii), (iii), a = 6f and a' = g~ for some f E HOmFG(V, M ) , g E HomFG(M, V). Hence the required assertion follows from Lemma l.lO(iv). (ii) First, we note that HomF(V, l G ) = V* and that the map
is an FG-isomorphism. Write v = a l ( l F ) for some a' E HomFH(lG,V). G I Then TT;(V) = T r H ( a )(IF) and so, by (i),
as required.
.
We now apply Corollary 1.11 to prove the following result.
Corollary 1.12. (Green (1983)). Let V be an FG-module and let H be a subgroup of G. Then the map
given by
.
(v E Inv(VH), a E Inv((V*)H) is a n F-bilinear form.
Proof. The last equality is Corollary l.ll(ii). This equality guarantees t h a t $H,V is well defined. Since '$'H,v is obviously F-bilinear, the result follows. We now return t o our G-algebra A and all the assumptions introduced a t the begining of this subsection. Of course, A can be regarded as an F G module in an obvious manner. Thus we may consider the bilinear form $P,A defined by Corollary 1.12.
2 The Broui5
- Robinson's theorem
367
With this information a t our disposal, we can now achieve the goal of this subsection which is to prove the following result.
Proposition 1.13.
(Broue' and Robinson (1986)). The bilinear forms
and $P,A : A:
x (A*)$+ F
have the same rank. Proof. Because ( A ,A) is a symmetric algebra, there is an F-space isomorphism 8 : A + A' given by
d(a)(b)=X(ab)
forall a , b ~ A
Since X is G-stable, 8 is obviously an isomorphism of FG-modules. In particular, 9 ( A P )= (A*)P. Now fix a', b1 E A'. Then, taking into account that X is symmetric, we have
and the result is established.
2
The Brou6
-
Robinson's theorem
Throughout this section, G denotes a finite group and F is an algebraically closed field of characteristic p > 0. We fix a finite-dimensional G-algebra A over F and a p-subgroup P of G. Suppose that X : A -t F is an F-linear map which is symmetric and G-stable. Then, by Lemma 1.3, there is a bilinear form
depending on P, G, A and A. This form is given by
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368
for a , b E A$ with b = ~ r $ ( b ' ) for some b' E A'. It turns out that we may replace P, G, A and X by 1, N G ( P ) / P , A ( P ) and Xp, respectively (for the definition of X p see Lemma 1.2(i)) such that the corresponding bilinear form A(P),A~
fi ,NG(P)/P
determines f:,$
.A(p)y(p)/p .
x A ( P ) ~ ( ~+ ) /F ~
in the sense that for for all a , b E A?,
(see Corollary 1.4). Of course, all of the above facts hold without the assumption that F is algebraically closed. AX We now probe more deeply into the nature of the form fpb. This will be achieved by replacing P , G, A and X by 1,N ( P , a ) , ~ n d ~ and k tr,, ) respectively, where N ( P , a ) , EndF(V,) and tr, are defined below. . denote by V, the associated simple APLet us fix a point a of A ~ We module, i.e. V, = A P e / ~ ( A P ) ewhere e E a (of course, the isomorphism class of V, is independent of the choice of e E a ) . We also let tr, : EndF(&)
-+
F
be the ordinary trace map. Let
be the representation of afforded by V,. Since F is algebraically closed, a, is surjective. We know that AP is an NG(P)/P-algebra in a natural way and we denote by N ( P , a ) the stabilizer of a in N G ( P ) / P , i.e.
We are now ready t o record the following observation.
Lemma 2.1. With the notation above, we have (i) EndF(V,) is a n N(P,a)-algebra via
(ii) The F-linear map tr, : EndF(V,) N ( P , a)-stable.
-t
F is both symmetric and
2 The Brou6
- Robinson's theorem
369
Proof. (i) Because I i e r o, is a maximal ideal of A' with a K e r a, and since each element of N ( P , a ) stabilizes a it follows from Proposition 3.1.2(iii) that h'era, is an N(P,a)-stable ideal of A'. Thus EndF(V,) becomes an N ( P , a)-algebra in a desired manner. (ii) It is obvious that tr, is symmetric. By Corollary 12.1.11 in Vol.1, the group N ( P , a ) acts on EndF(V,) via inner automorphisms. Hence tr, must be also N ( P , a)-stable. 1 Owing t o Lemmas 1.3 and 2.1, we may now define the bilinear form
with the role of P , G , A and X played by 1, N ( P , a ) , EndF&) and tr,, respectively. with f El,N(P,a) n d F ( V a ) ' t T ~This will be Our next task is to tie together achieved with the aid of the following result.
f[$
Proposition 2.2.
(Broue' and Robinson (1986)). Assume that J(A)
Iier X
Then
(i) For all a, b E A s , we have
where S denotes a set of all representatives for the NG(P)/P-orbits of local points of A' such that X(a) # 0. (ii) We have
Proof. (i) Owing t o Lemma 3.2.6(i), the Brauer morphism
induces a bijection from the set of all local points of A ' onto the set of all points of A(P). Hence, by Corollary 1.4, we are reduced to the case where P = 1 (replacing A, G, P, X by A ( P ) , N G ( P ) / P , 1, Xp, respectively). In that
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370
case G ( a ) = N ( l , a ) is the stabilizer of a in G, Let b = TrF(bl) be an element of A? and, for each a E pt(Af), let T ( o ) be a. right transversal for G ( a ) in G. Then, for any given a E pt(Af), it follows from Lemma 1.2.6 (with I< = G, D = G ( a ) and H = 1) that
Hence, by (I), we have
By the definition of the form f$, we then have to prove that for all a E A?,
that is aES
By Lemma 1.6, we know that
On the other hand, for any given a E pt(A), we have
2
The Brou6
- Robinson's
theorem
This establishes (2), by applying (3) and (4). (ii) Again, we may reduce t o the case P = 1, and we do so. If
is such that tr,(v$)
= 0 for all $ E EndF(V,), then 9 = 0. Thus
Furthermore, the map
.
is clearly surjective. The desired conclusion is therefore a consequence of (i), (5) and Lemma 1.1. We now concentrate on the special case where A is a primitive G-algebra, i.e. lAis a primitive idempotent of A ~ Recall . that a defect pointed group of A is defined to be a defect pointed group of G,, where cr = {IA). Thus, if P, is a defect pointed group of A , then P is a defect group of A (i.e. P is a defect group of {IA))and y is a local point of A' such that
(see Theorem 3.4.2). Note also that, by Theorem 3.4.2, any other local point of AP is in the NG(P)/P-orbit of y. Here we have used the obvious fact that, for any pointed group Qp on A, Qp C G{l,). T h e point y above is called a source point of A (or of GI1,)) corresponding t o P. In what follows, P denotes a psubgroup of G. We are now ready t o prove our main result, which provides a detailed analysis of the form ~ A'A P , G: A$ X
where X : A
+
A$
+F
F is an F-linear map which is symmetric and G-stable.
Theorem 2.3. (Broue' and Robinson (1986)). Assume that A is a primitive G-algebra and that J ( A ) C K e r A. Then (i) If f g g # 0, then P is contained in a defect group of A. (ii) If P is a defect group of A and y is a source point of A corresponding to P, then f i g # 0 if and only if A(7) # 0
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372
(iii) If P is a defect group of A and y is a source point of A corresponding to P with X(y) # 0, then
where N ( P , y) is the stabilizer of y in N G ( P ) / P . (iv) If AG 2 Z(A), then f:$ # 0 if and only if P is a defect gmup of A and X(y) # 0 where y is a source point of A corresponding to P . C Z(A), P is a defect group of A and A(y) # 0 where y is a (v) If source point of A corresponding to P, then
Proof. (i) Let D be a defect group of A. Owing t o Corollary 1.2.8(i), we have 1E C A,PDg-'np
AB C
g€G
Hence, if P is not G-conjugate t o a subgroup of D , i.e. if all gDg-l n P are proper subgroups of P, then by Proposition 5.5.3 in Vol.1, 1 E A; for some proper subgroup H of P. But then A ( P ) = 0 and so, by Corollary 1.4, --
f
A,X
= (ii) and (iii) Assume that P is a defect group of A and let y be a source point of A corresponding to P. Then, by Theorem 3.4.2, there exists exactly one Nc(P)/P-orbit of local points of A', namely the orbit of y. Hence, by Proposition 2.2(ii), P,G
and, in particular, iff:,$ # 0 then X(y) # 0. Conversely, suppose that A(?) # 0. Then, by Theorem 4.8(ii) (with A(H,) replaced by ~ ~ ( and a BT; ) by nH), 0 we have
for all x E AP . y .AP. But 1 E A;, so we may take x such that T T ~ ( X=) 1. We conclude therefore that
3 Applications to the group algebra
proving t h a t f($ # 0. (iv) and (v) Suppose that AG C Z(A). By (i) and (ii), t o prove (iv), it suffices t o show that if P is strictly contained in a defect group of A, then f i i = 0. But if P is strictly contained in a defect group of A, then 1 9 and so A: g J ( A ~ )
AS
since A ' is local (by Lemma 1.3.4(ii)). It follows that each element of A: is central and nilpotent. Using the fact that X vanishes on each nilpotent we therefore deduce that element (Lemma 1.6) and the definition of f?:,
f$
= 0.
Because A ' is central in A, it is mapped into the center of EndF(V,) by the homomorphism a,. Hence a , ( ~ g )= 0 or F and, on the other hand,
by virtue of (ii) and Proposition 2.2(i). Thus o,(A$) = F and so, by Proposition 2.2(i), rank( =1
fgi)
proving (v).
3
Applications to the group algebra
In this section, G denotes a finite group and R a complete discrete valuation ring of characteristic 0 such that the residue field F = R / J ( R ) is algebraically closed of prime characteristic p. For each x = CgEG xgg E RG, xg E R , let Cgg
i?=
((z = xg
s EG Then the map
{
+ J(R))
RG + F G x - 3
is a surjective homomorphism with kernel J ( R ) G . For any R-linear map $ : R G -, R, we denote by 4 the reduction of $ modulo J ( R ) , i.e. 4 : F G + F is the F-linear map given by 4 ( 2 ) = $(z) for all x E R G
Bilinear Forms on G-Algebras
From now on, we fix a n F-linear map
which is a class function on G. Of course, regarding F G as a G-algebra is a natural way, X is both symmetric and G-stable. In what follows, we also assume that J(FG) Iier X Our point of departure is the following elementary result. Lemma 3.1. There exists a n R-linear combination of characters of where is the reduction of $ modulo R-free RG-modules such that X =
4,
4
J/I
J(R).
Proof. Owing t o Lemma 1.6, X is an F-linear combination of the characters of simple FG-modules. Since F G E R G / J ( R ) G , the required assertion follows. Let e be a block idempotent of RG with defect group P. Assume that x is a n R-linear combination of characters of RG-modules. Owing t o Theorem 5.8.3, i2 is a block idempotent of F G with defect group P. Because the restriction of ji to F G e (which we also denote by j j ) is symmetric and Gstable, we may introduce the bilinear form
) ~Z(FGE) is local, we may define a source point y Moreover, since ( F G ~ = of FGE corresponding to P. Lemma 3.2. With the assumptions and notation above, the following conditions are equivalent : (i) x(e)/(G : P ) is invertible in R.
Proof. It is plain that J ( F G ) K e r X. Because P is a defect group of E, it follows from Theorem 2.3(ii) that (ii) is equivalent to (iii).
3 Applications to the group algebra
375
Next we show that (i) is equivalent to (iii). To this end, we first write e = ~ r g ( x ) for some x E (RG~)' = ( R G ) ' ~ Then we must have x(e> = (G : P ) x ( x ) and
FGZ % - fP,G ' (e, e) = X ( 2 ) = x(x)
Owing t o (I), condition (i) is equivalent t o the requirement that ~ ( x #) 0. Thus, by (2), it suffices t o verify that
Bearing in mind that
and
.
~[~'"(J(Z(FG)Z), x) = 0 for all x E Z(FGE), the result follows.
As a n application, we now prove the following result which illustrates the significance of the introduced bilinear form. Theorem 3.3. (Broue' and Robinson (1986)). Let x be a n R-linear combination of characters of RG-modules, let P be a p-subgroup of G and let e l , . . . ,en be all block idempotents e of RG with defect group P and such that x(e)/(G : P ) is invertible in R
Then we have
FG,%
rank(fp,c ) = n Proof. Let e l , ez, . . . , e n , e n + l , . . . , e m be all block idempotents of RG. Then
rank
(f:z"
=
a i=l
( f x )
(by Lemma 1.1)
Bilinear Forms on G-Algebras
=
rank i=l
(f:ga'fi)
(by Lemma 3.2)
= n(by Lemma 3.2 and Theorem 2.3(v))
as we wished t o show. H In view of the above result, it is important t o provide x for which x(e)/(G : P ) is invertible in R. This is achieved by the lemma below. The proof depends on Theorem 1.14.6 which guarantees that multiplication by e preserves ( R G ~ ~ ) ~ .
Lemma 3.4. Let Q be a Sylow p-subgroup of G and let x be the character of the RG-module ( I ~ ) If~ e. is a block idernpotent of R G with defect group P, then x(e)/(G : P ) is invertible in R
Proof. Owing to Lemma 3.2, it suffices t o show that
e g g with eg E R so that e = CsEG e g g . Because To this end, write e = CSEG B ~ $ ~ (#E0) (Lemma 1.12.13), there exists a p-regular element t of C G ( P ) such that %-I # 0. Bearing in mind that
it therefore suffices to show that
Denote by RG,I the R-linear span of the set Gpl of all pregular elements of G . Then, by Theorem 1.14.6, multiplication by e preserves ( R G , , ) ~ . Thus
which clearly implies (3). H
3 Applications to the group algebra
377
As an application, we finally prove
Corollary 3.5. (Broue' and Robinson (1986)). Let P be a p-subgroup ~ , Q is a Sylow of G , and let x be the character of the RG-module ( I ~ )where p-subgroup of G. Then the rank of the form
is equal to the number of blocks of FG with defect group P .
Proof. Apply Theorem 3.3 and Lemma 3.4. H We shall return t o the subject of counting blocks of group algebras in our future treatment of block theory.
Bilinear Forms on G- Algebras We have now come t o the last leg of our long journey into the theory of Galgebras. This chapter contains some basic information concerning bilinear forms on G-algebras. Here we concentrate exclusively on the case where the ground coefficient ring is a field of prime characteristic p. The bilinear forms that we consider below are far from being nonsingular. The case where the ground coefficient ring is a complete discrete valuation ring requires a different technique which can be found in a work of Thdvenaz (1988a). In the first section, we develop some general properties of bilinear forms on G-algebras with an eye to future applications t o group algebras. It turns out that the most convenient setting t o work with is that of G-algebras endowed with G-stable bilinear forms. These forms are somewhat analogous t o that introduced by Green (1983). The main results are contained in Sec.2. In Sec.3 we concentrate exclusively on the group algebra case and introduce a bilinear form whose rank is equal t o the number of blocks of F G with defect group P. Here F is a field of characteristic p > 0 and P is a given p-subgroup of G. T h e chapter also contains some additional results which are of independent interest. To facilitate the reading, important definitions have been repeated where necessary.
Bilinear Forms on G-Algebras
1
Preliminary results
In what follows, F denotes a field and all vector spaces over F are assumed t o be finite-dimensional. From Lemma 1.2 onwards, it will be assumed that charF = p > 0. For convenience of exposition, the section is divided into three subsections.
A. General facts Our aim here is to provide some general information concerning bilinear forms on G-algebras. We also introduced a specific form that will be studied in the next section. Let V be a vector space over F and let
be a symmetric bilinear form. Then the rank of f , written r a n k ( f ) is defined by rank( f ) = d i r n F ( v / v L ) where
v L ={x E V l f ( x , V ) =
0)
It is clear that the rank of f is equal to the rank of the matrix
where a;j = f(vi,vj) and vl,. . . ,v, is a basis of V . Our point of departure is the following elementary lemma concerning the rank of symmetric bilinear forms.
Lemma 1.1. Let f : V x V -+ F and let f; : V, x V, -+ F be symmetric bilinear forms, 1 i i n . Suppose that for each i E (1,. . . ,n ) , there is an F-linear map n; : V + V, and a nonzero pi E F such that the following two properties hold : (i) The induced map
<
1 Preliminary results
is surjective.
(ii) f (x, Y)= Cy=l~ i f i ( n i ( x )ri(Y)) , Then we have n
for all X ,Y E V
Proof. We may harmlessly assume that each pi = 1. Consider the symmetric bilinear form
defined by
n
Then, owing t o (ii), we have
It therefore follows from (i) that, for any given x E V,
Hence r induces an F-isomorphism V/VL
as desired.
--+
(ny=&)/(nE1
K)'. Thus
.
From now on, we fix a finite group G and a finite-dimensional G-algebra A over a field F of characteristic p > 0. We also assume that A is endowed with an F-linear map X:A-+F which is symmetric in the sense that X(ab) = X(ba) for all a , b E A and also G-stable in the sense that
Bilinear Forms on G-Algebras
356
If A is an interior G-algebra, then as X we can always choose the character of a matrix representation of A over F. In what follows, P denotes a p-subgroup of G and Br$ : AP
--,
A(P)
is the corresponding Brauer morphism. We remind the reader that, by Lemma 1.7.1, both AP and A(P) are NG(P)/P-algebras and Br$ is a homomorphism of NG(P)/P-algebras over F. The following observation will allow us to introduce a bilinear form associated with P, G, A and A.
Lemma 1.2. The following properties hold : (i) lier(Br;) C I i e r A. (ii) X induces an F-linear map Xp : A(P) -t F given by Xp(~r;(a)) = X(a)
for all a E AP
(iii) Xp is symmetric and Nc(P)/P-stable.
Proof. It is clear that (ii) and (iii) follow from (i). To prove (i), we first note that, by definition
where, by convention, for P = 1 we have l i e r ( ~ r ; ) = 0. Assume that & is a proper subgroup of P and let a E A;. We must show that X(a) = 0. Write a = ~ r ; ( x ) for some s E AQ and let T be a left transversal for Q in P. Then, since X is G-stable, we have
as required. Following Brouk and Robinson (1986), we now define the bilinear form
""
~ P , G: A$
x A$
--,
F
as follows : if a, b E A$ with b = ~ r g ( b ' )for some b' E A ~ then ,
1 Preliminary results
357
Our next aim is t o record some elementary properties of the introduced bilinear form. AX Lemma 1.3. The form fp;G is well defined, symmetric, and associa-
tive. Proof. Assume that a = ~ r g ( a ' )and b = Tr?(b1) for some a', b' E A'. Let the F-linear map Xp : A(P) -+ F be defined as in Lemma 1.2(ii). Then
Observe also that, by Lemma 1.7.2(i),
Thus we must also have Br$(a) = Tr,N"'P'lP
(B$ (a'))
and Br;(b)
T$ = Tr,N ~ ( P ) / P ( ~ (bl))
(2)
Replacing G, P, A by N G ( P ) / P , 1, A(P), we may therefore assume that P = 1. Because X is G-stable, we have X(g(al)b) = /\(a' 9-'b) for all g E G. Thus
Consequently, if Trf(bl) = Tr?(bt1), then X(abl) = X(abl1) and so the form is well defined. Since X(abf) = X(bfa), the form is symmetric by virtue of (3). Moreover, because cb = ~ r f ( c b I )for all c E AG, we have f;lih(ac, b) = X(acbl) = f s ( a , cb) This shows that the form is also associative, as required. B Owing t o Lemmas 1.2 and 1.3, we may now define the bilinear form
Bilinear Forms on G-Algebras
358
with the role of P, G, A and X played by 1, N G ( P ) / P , A ( P ) and X p , respectively. The following observation explains how the new bilinear form can be brought into argument. For all a , b E A:, we have
Corollary 1.4.
Proof. Write b = ~ r $ ? ( b ' )for some b' E A ~ Then . A(P),XP
f,,,(p,,p(~rf
(a), ~ r f ( b ) ) = ~ p ( ~ r S ( a ) ~ r S ( b ' ) )(by (2)) = Xp(~r$(abl)) = X(abl) (by Lemma 1.2(ii))
= f;$(a,b) as required. W Next we record the following elementary property. Lemma 1.5.
For any integer n
> 1, the ordinary trace map
is the only (up to scalar multiples) symmetric F-linear map from M n ( F ) to the field F. Proof. Let V be the F-space of all symmetric F-linear maps M,(F)
+
F . We must show that dimFV = 1. But, for any q E V ,
and hence the result.
.
dim^ Mn ( F )/ [ M n ( F ) Mn (F)] = 1,
For any point a of A, let X , denote the character of A afforded by the corresponding simple A-module Ae/J(A)e where e E a. Of course, X , is independent of the choice of e . We let X(a) denote X(e) (which depends
1 Preliminary results
359
only on a , not on the particular e chosen). In what follows, pt(A) denotes the set of all points of A. Lemma 1.6. Assume that F is a splitting field for A and that J ( A )
Ker A. Then
C
A=
c
X(a)xa
aEpt(A)
, A(a) be the unique block of A/J(A) Proof. Given a E P ~ ( A ) let which contains the point a J ( A ) of A/ J(A). Then, by Proposition 3.1.2,
+
is a block decomposition of A/J(A). On the other hand, since F is a splitting field for A, it follows from Corollary 11.1.4 in Vol.1 that, for any given a E pt(A), A(a) Mn, (F)
>
for some integer n, 1. Because J ( A ) 2 Ker X and, by Lemma 1.5, there is (up to scalar multiples) only one symmetric F-linear map Mn,(F) -+ F, we may write
It is easy t o see that x,(,f3) = Sap for all a , p E pt(A), Thus pa = X(a) for each a E pt(A), as required. W
B. The group algebra case To break the monotony, we now offer an important example pertaining t o the G-algebra F G where G acts on F G by conjugation. We remind the reader that F is a field of characteristic p > 0 and P is a p-subgroup of G. For any conjugacy class C of G, S(C) denotes a defect group of C (with respect t o p). For any subset S of G, S+ E F G denotes the sum of all elements in S. If HI, Hz are subgroups of G, then we write H1 GG H2 to indicate that H1 is G-conjugate to a subgroup of Hz. Of course, HI CG H2 means that H1 is G-conjugate to a proper subgroup of H2. Let C1,. . . ,Cn be all conjugacy classes of G with S(C;) CG P, where P is a given p-subgroup of G. Then, by Corollary 1.6.5,
Bilinear Forms on G-Algebras
360
form a basis of ( F G ) ~Keeping . the above notation, we now record
Example 1.7. Let X : FG + F be defined by X ( ~ , ~ ~ = x X~I , g ) x, E F . Then (i) X is an F-linear map which is symmetric and G-stable. Moreover, for any subsets X , Y of G ,
(ii) For any X , Y E {C1,.. . ,C,), we have FG,X
f p , ~( X t
,Y + )=
I X l p ~ / l G l p ~ if
y = x-' and 16(X)I = /PI otherwise
(iii) ((FG):)' = l i e r ( ~ r Fn~( )F G ) = ~ V where V is the F-linear span of all C t , where C is a conjvgacy class of G with 6 ( C ) C G P. Proof. (i) It is clear that X is an F-linear map which is symmetric and G-stable. By definition of A , we also have
where X(xy) = 1 if x = y-'
and X(xy) = 0 if x f y-'.
Thus
as required. (ii) Choose g E Y and a Sylow psubgroup Q of CG(g)such t h a t Q G P. Let C be the P-conjugacy class of g. Then
TT;(C') = ( C G ( S ): C p ( g ) ) Y t and (CG(g): C P ( g ) )is a p'-number since Q C Cp(g). Hence
and therefore
1 Preliminary results
361
# X-',
then X n C-*= 0 and so f pFGG' X (x+ , Y + ) = 0. Hence we may assume that Y = X-', in which case X n C-' = C-l. If 16(X)I < [ P I , then FG X p divides IC-ll = ( P : Cp(g)). Hence, in this case, fP,G' ( X f ,Y t ) = 0. Finally, assume that 16(X)I = [PI. Then Q = P and so
If Y
which forces
as required. (iii) Given a nonzero z in ( F G ) ~ , write x = X I Xt~ - - . XrX: for some nonzero A; E F and some XiE {C*,.. . ,C,), 1 5 i 5 r . Then, by if and only if IS(X;)I < [PI for all 1 5 i 5 r (which (ii), x E ((FG)?)' means that x E V). The latter is equivalent to the requirement that each P-conjugacy class in X i , 1 5 i 5 r , is nontrivial, i.e. t o x E K e r ( ~ r ; ~ ) (see Lemma 1.12.9).
+
Keeping the notation of the preceding result, we now record
Proposition 1.8. Let e be any central idempotent of F G . Then the rank of the bilinear form
is given by
Proof. P u t V = (FG)?~. Since (FG): it is clear that V (FG)?. We claim that
is an ideal of ( F G ) ~= Z ( F G ) ,
Since
V nl i e r ( ~ r ; ~ )
c
(FG)$ n l i e r ( ~ r F ~ )
= ((FG>~>'
(by Example 1.7(iii))
Bilinear Forms on G-Algebras
362
it is clear that V fl ~ i e r ( ~ r 2 F v' ~. ) Conversely, let xe E V' for some z E (FG):.
Then, for any y E (FG)?,
since, by Lemma 1.3, f[gt"s associative. Thus, by Example 1.7(iii), r e E ~ e r ( B r ; ~ which ) proves (4). We now deduce that
.
where the last equality is a consequence of Lemma 1.7.2(ii) and our identification of ( F G ) ( P ) with F C G ( P ) (see Corollary 1.7.13). We close this subsection by recording the following consequence of the above result.
Corollary 1.9.
The bilinear forms
have the same rank.
Proof. Setting H = NG(P), we obviously have N G ( P ) = N H ( P ) and C G ( P ) = CH(P). Applying Proposition 1.8 twice (with e = l ) , we have
as desired. H
1 Preliminary results
363
C. Comparison with a bilinear form of Green
Again, we assume that A is a finite-dimensional G-algebra over a field F of characteristic p > 0. We also assume that A is endowed with an F-linear map X:A+F which is symmetric and G-stable. In addition, we assume that (A, A) is a symmetric algebra, i.e. we require that K e r X contains no nonzero left (equivalently, right) ideal. As before, P denotes a p-subgroup of G. Our aim is t o show that the rank of the bilinear form
introduced prior to Lemma 1.3 is equal to the rank of a bilinear form introduced in a work of Green (1983) (see Corollary 1.12 with H = P). All FG-modules below are assumed to be finitely generated. Given an ~)) FG-module V and a subgroup H of G, we put V$ = ~ r g ( ~ n v ( Vwhere
is the relative trace map. Recall that Inv(VH) denotes the subspace of all H-invariant elements of V , while Inv(V) is the subspace of all G-invariant elements of V. As usual V* = HomF(V, F) is the contragredient to V. Thus V* is an FG-module via (g f)(v) = f(g-'v) for all v E V, g E G, f E V*. For the purpose of this subsection, we need only to prove that
for all v E Inv(VH), cr E Inv((V*)H). We shall derive this property as a consequence of a more general result that will be needed later. Let U and V be FG-modules. Recall that HomF(U, V ) is an FG-module via (gO)(u) = g(O(g-lu))
for all
u
E U,g E G,O E H o r n ~ ( UV) ,
Then HomFH(U, V) is the space of all H-invariant elements of HomF(U, V), HomFG(U, V) is the space of all G-invariant elements of HomF(U, V) and
Bilinear Forms on G-Algebras
364
Of course, ~ r Z ( ~ o r Vn )~) is~a (subspace ~ , of H o m F G ( U , V ) . Let l H be the trivial FH-module, i.e. l H = F and H acts trivially on F, and let M = ( 1 ~ ) Then ~ . M is a transitive permutation FG-module which has an F-basis { u , ~ x E G I H ) ( G / H is the set of all left cosets of H in G) on which G acts by
gu, = ugX
for all g E G, x E G/H
The basic element u = U H is H-invariant and M = F G . u. In what follows, F = 1~ is the trivial FG-module. Consider the maps
t E H o ~ F G ( MI G, ) , 6 E H o ~ F H ( M1 , ~
)
defined as follows :
( c ,cx E F ) . We now fix an FG-module V and prove the following result. Lemma 1.10. Furhter to the notion above, put E = E n d F G ( M ) . Then (i) There is a homomorphism p : E -+ F of F-algebras such that
where 8 E E and 8 ( u ) = ~ x E G I HcxuX. Moreover, 87 = p(O)q, ( 8 = p(O)( for any 0 E E . (ii) The map HonzFG(V,d4)-+ H o m F H ( V ,l G ) ,f H 6f is an F-isomorphisr (iii) The nzap HonzFc(M,V) + H o m F H ( l G , V ) ,g H gE is an F isomorphism. (iv) Given f E HomFG(V,M ) , g E H o m ~ ~ ( ?vd ), ,put a = 6 f, a' = ge, P = t f and p' = gq. Then (a) a P ' ( 1 ~=) P a ' ( 1 ~ ) . (b) p = T r $ ( a ) ,/3' = T r E ( o 1 ) .
1 Preliminary results
365
Proof. (i) It is clear that the spaces HOmFG(lG,M), HomFG(M, 1 ~ ) are, respectively, left and right E-modules (acting by composition of maps). Moreover, these spaces are one-dimensional; in fact HomFG(lG,M ) = F . q and HomFG(M, l G ) = F . (. Therefore there are homomorphisms p , p' : E + F of F-algebras such that 0q = p(Q)q, t 0 = pl(0)<
for all 0 E E
(1)
We claim that p = p'. Indeed, we first check that 6q(lF) = 1~ = l ~ ( 1 ~ ) . Then from (1) we obtain
Finally, a direct calculation shows that SOq(lF) = <0&(lF) = cz, where O(u) = C c,u, (observe that an element 0 E E is determined by 0(u), since M = F G . u). This implies (i). (ii) and (iii) This is a special case of Proposition 18.1.6 in VoI.1. (iv) From (1) (putting B = f g ) , we obtain a string of equalities : p(fg)
= 6f977(1~)= aD'(lF)
= t f g ~ ( 1= ~ )D a l ( l ~ ) proving (a). It is a consequence of the definitions of 7 , E, [ and 6 that TT$(S) = <, T T ~ ( E=) q. Hence
and Dl
= gq = ~ T GT ~ (=ET) TG ~ ( ~=ETT$(CY') )
proving (b). W As an application, we now record
Corollary 1.11. Let V be an FG-module and let H be a subgroup of G. Then (i) POTany a E H o ~ F H ( V l, ~ ) a', E H o m ~ ~ ( lV), c,
Bilinear Forms on G-Algebras
366
(ii) For any v E Inv(VH), a E Inv((V8)H), we have
Proof. (i) By Lemma l.lO(ii), (iii), a = 6f and a' = g~ for some f E HOmFG(V, M ) , g E HomFG(M, V). Hence the required assertion follows from Lemma l.lO(iv). (ii) First, we note that HomF(V, l G ) = V* and that the map
is an FG-isomorphism. Write v = a l ( l F ) for some a' E HomFH(lG,V). G I Then TT;(V) = T r H ( a )(IF) and so, by (i),
as required.
.
We now apply Corollary 1.11 to prove the following result.
Corollary 1.12. (Green (1983)). Let V be an FG-module and let H be a subgroup of G. Then the map
given by
.
(v E Inv(VH), a E Inv((V*)H) is a n F-bilinear form.
Proof. The last equality is Corollary l.ll(ii). This equality guarantees t h a t $H,V is well defined. Since '$'H,v is obviously F-bilinear, the result follows. We now return t o our G-algebra A and all the assumptions introduced a t the begining of this subsection. Of course, A can be regarded as an F G module in an obvious manner. Thus we may consider the bilinear form $P,A defined by Corollary 1.12.
2 The Broui5
- Robinson's theorem
367
With this information a t our disposal, we can now achieve the goal of this subsection which is to prove the following result.
Proposition 1.13.
(Broue' and Robinson (1986)). The bilinear forms
and $P,A : A:
x (A*)$+ F
have the same rank. Proof. Because ( A ,A) is a symmetric algebra, there is an F-space isomorphism 8 : A + A' given by
d(a)(b)=X(ab)
forall a , b ~ A
Since X is G-stable, 8 is obviously an isomorphism of FG-modules. In particular, 9 ( A P )= (A*)P. Now fix a', b1 E A'. Then, taking into account that X is symmetric, we have
and the result is established.
2
The Brou6
-
Robinson's theorem
Throughout this section, G denotes a finite group and F is an algebraically closed field of characteristic p > 0. We fix a finite-dimensional G-algebra A over F and a p-subgroup P of G. Suppose that X : A -t F is an F-linear map which is symmetric and G-stable. Then, by Lemma 1.3, there is a bilinear form
depending on P, G, A and A. This form is given by
Bilinear Forms on G-Algebras
368
for a , b E A$ with b = ~ r $ ( b ' ) for some b' E A'. It turns out that we may replace P, G, A and X by 1, N G ( P ) / P , A ( P ) and Xp, respectively (for the definition of X p see Lemma 1.2(i)) such that the corresponding bilinear form A(P),A~
fi ,NG(P)/P
determines f:,$
.A(p)y(p)/p .
x A ( P ) ~ ( ~+ ) /F ~
in the sense that for for all a , b E A?,
(see Corollary 1.4). Of course, all of the above facts hold without the assumption that F is algebraically closed. AX We now probe more deeply into the nature of the form fpb. This will be achieved by replacing P , G, A and X by 1,N ( P , a ) , ~ n d ~ and k tr,, ) respectively, where N ( P , a ) , EndF(V,) and tr, are defined below. . denote by V, the associated simple APLet us fix a point a of A ~ We module, i.e. V, = A P e / ~ ( A P ) ewhere e E a (of course, the isomorphism class of V, is independent of the choice of e E a ) . We also let tr, : EndF(&)
-+
F
be the ordinary trace map. Let
be the representation of afforded by V,. Since F is algebraically closed, a, is surjective. We know that AP is an NG(P)/P-algebra in a natural way and we denote by N ( P , a ) the stabilizer of a in N G ( P ) / P , i.e.
We are now ready t o record the following observation.
Lemma 2.1. With the notation above, we have (i) EndF(V,) is a n N(P,a)-algebra via
(ii) The F-linear map tr, : EndF(V,) N ( P , a)-stable.
-t
F is both symmetric and
2 The Brou6
- Robinson's theorem
369
Proof. (i) Because I i e r o, is a maximal ideal of A' with a K e r a, and since each element of N ( P , a ) stabilizes a it follows from Proposition 3.1.2(iii) that h'era, is an N(P,a)-stable ideal of A'. Thus EndF(V,) becomes an N ( P , a)-algebra in a desired manner. (ii) It is obvious that tr, is symmetric. By Corollary 12.1.11 in Vol.1, the group N ( P , a ) acts on EndF(V,) via inner automorphisms. Hence tr, must be also N ( P , a)-stable. 1 Owing t o Lemmas 1.3 and 2.1, we may now define the bilinear form
with the role of P , G , A and X played by 1, N ( P , a ) , EndF&) and tr,, respectively. with f El,N(P,a) n d F ( V a ) ' t T ~This will be Our next task is to tie together achieved with the aid of the following result.
f[$
Proposition 2.2.
(Broue' and Robinson (1986)). Assume that J(A)
Iier X
Then
(i) For all a, b E A s , we have
where S denotes a set of all representatives for the NG(P)/P-orbits of local points of A' such that X(a) # 0. (ii) We have
Proof. (i) Owing t o Lemma 3.2.6(i), the Brauer morphism
induces a bijection from the set of all local points of A ' onto the set of all points of A(P). Hence, by Corollary 1.4, we are reduced to the case where P = 1 (replacing A, G, P, X by A ( P ) , N G ( P ) / P , 1, Xp, respectively). In that
Bilinear Forms on G-Algebras
370
case G ( a ) = N ( l , a ) is the stabilizer of a in G, Let b = TrF(bl) be an element of A? and, for each a E pt(Af), let T ( o ) be a. right transversal for G ( a ) in G. Then, for any given a E pt(Af), it follows from Lemma 1.2.6 (with I< = G, D = G ( a ) and H = 1) that
Hence, by (I), we have
By the definition of the form f$, we then have to prove that for all a E A?,
that is aES
By Lemma 1.6, we know that
On the other hand, for any given a E pt(A), we have
2
The Brou6
- Robinson's
theorem
This establishes (2), by applying (3) and (4). (ii) Again, we may reduce t o the case P = 1, and we do so. If
is such that tr,(v$)
= 0 for all $ E EndF(V,), then 9 = 0. Thus
Furthermore, the map
.
is clearly surjective. The desired conclusion is therefore a consequence of (i), (5) and Lemma 1.1. We now concentrate on the special case where A is a primitive G-algebra, i.e. lAis a primitive idempotent of A ~ Recall . that a defect pointed group of A is defined to be a defect pointed group of G,, where cr = {IA). Thus, if P, is a defect pointed group of A , then P is a defect group of A (i.e. P is a defect group of {IA))and y is a local point of A' such that
(see Theorem 3.4.2). Note also that, by Theorem 3.4.2, any other local point of AP is in the NG(P)/P-orbit of y. Here we have used the obvious fact that, for any pointed group Qp on A, Qp C G{l,). T h e point y above is called a source point of A (or of GI1,)) corresponding t o P. In what follows, P denotes a psubgroup of G. We are now ready t o prove our main result, which provides a detailed analysis of the form ~ A'A P , G: A$ X
where X : A
+
A$
+F
F is an F-linear map which is symmetric and G-stable.
Theorem 2.3. (Broue' and Robinson (1986)). Assume that A is a primitive G-algebra and that J ( A ) C K e r A. Then (i) If f g g # 0, then P is contained in a defect group of A. (ii) If P is a defect group of A and y is a source point of A corresponding to P, then f i g # 0 if and only if A(7) # 0
Bilinear Forms on G-Algebras
372
(iii) If P is a defect group of A and y is a source point of A corresponding to P with X(y) # 0, then
where N ( P , y) is the stabilizer of y in N G ( P ) / P . (iv) If AG 2 Z(A), then f:$ # 0 if and only if P is a defect gmup of A and X(y) # 0 where y is a source point of A corresponding to P . C Z(A), P is a defect group of A and A(y) # 0 where y is a (v) If source point of A corresponding to P, then
Proof. (i) Let D be a defect group of A. Owing t o Corollary 1.2.8(i), we have 1E C A,PDg-'np
AB C
g€G
Hence, if P is not G-conjugate t o a subgroup of D , i.e. if all gDg-l n P are proper subgroups of P, then by Proposition 5.5.3 in Vol.1, 1 E A; for some proper subgroup H of P. But then A ( P ) = 0 and so, by Corollary 1.4, --
f
A,X
= (ii) and (iii) Assume that P is a defect group of A and let y be a source point of A corresponding to P. Then, by Theorem 3.4.2, there exists exactly one Nc(P)/P-orbit of local points of A', namely the orbit of y. Hence, by Proposition 2.2(ii), P,G
and, in particular, iff:,$ # 0 then X(y) # 0. Conversely, suppose that A(?) # 0. Then, by Theorem 4.8(ii) (with A(H,) replaced by ~ ~ ( and a BT; ) by nH), 0 we have
for all x E AP . y .AP. But 1 E A;, so we may take x such that T T ~ ( X=) 1. We conclude therefore that
3 Applications to the group algebra
proving t h a t f($ # 0. (iv) and (v) Suppose that AG C Z(A). By (i) and (ii), t o prove (iv), it suffices t o show that if P is strictly contained in a defect group of A, then f i i = 0. But if P is strictly contained in a defect group of A, then 1 9 and so A: g J ( A ~ )
AS
since A ' is local (by Lemma 1.3.4(ii)). It follows that each element of A: is central and nilpotent. Using the fact that X vanishes on each nilpotent we therefore deduce that element (Lemma 1.6) and the definition of f?:,
f$
= 0.
Because A ' is central in A, it is mapped into the center of EndF(V,) by the homomorphism a,. Hence a , ( ~ g )= 0 or F and, on the other hand,
by virtue of (ii) and Proposition 2.2(i). Thus o,(A$) = F and so, by Proposition 2.2(i), rank( =1
fgi)
proving (v).
3
Applications to the group algebra
In this section, G denotes a finite group and R a complete discrete valuation ring of characteristic 0 such that the residue field F = R / J ( R ) is algebraically closed of prime characteristic p. For each x = CgEG xgg E RG, xg E R , let Cgg
i?=
((z = xg
s EG Then the map
{
+ J(R))
RG + F G x - 3
is a surjective homomorphism with kernel J ( R ) G . For any R-linear map $ : R G -, R, we denote by 4 the reduction of $ modulo J ( R ) , i.e. 4 : F G + F is the F-linear map given by 4 ( 2 ) = $(z) for all x E R G
Bilinear Forms on G-Algebras
From now on, we fix a n F-linear map
which is a class function on G. Of course, regarding F G as a G-algebra is a natural way, X is both symmetric and G-stable. In what follows, we also assume that J(FG) Iier X Our point of departure is the following elementary result. Lemma 3.1. There exists a n R-linear combination of characters of where is the reduction of $ modulo R-free RG-modules such that X =
4,
4
J/I
J(R).
Proof. Owing t o Lemma 1.6, X is an F-linear combination of the characters of simple FG-modules. Since F G E R G / J ( R ) G , the required assertion follows. Let e be a block idempotent of RG with defect group P. Assume that x is a n R-linear combination of characters of RG-modules. Owing t o Theorem 5.8.3, i2 is a block idempotent of F G with defect group P. Because the restriction of ji to F G e (which we also denote by j j ) is symmetric and Gstable, we may introduce the bilinear form
) ~Z(FGE) is local, we may define a source point y Moreover, since ( F G ~ = of FGE corresponding to P. Lemma 3.2. With the assumptions and notation above, the following conditions are equivalent : (i) x(e)/(G : P ) is invertible in R.
Proof. It is plain that J ( F G ) K e r X. Because P is a defect group of E, it follows from Theorem 2.3(ii) that (ii) is equivalent to (iii).
3 Applications to the group algebra
375
Next we show that (i) is equivalent to (iii). To this end, we first write e = ~ r g ( x ) for some x E (RG~)' = ( R G ) ' ~ Then we must have x(e> = (G : P ) x ( x ) and
FGZ % - fP,G ' (e, e) = X ( 2 ) = x(x)
Owing t o (I), condition (i) is equivalent t o the requirement that ~ ( x #) 0. Thus, by (2), it suffices t o verify that
Bearing in mind that
and
.
~[~'"(J(Z(FG)Z), x) = 0 for all x E Z(FGE), the result follows.
As a n application, we now prove the following result which illustrates the significance of the introduced bilinear form. Theorem 3.3. (Broue' and Robinson (1986)). Let x be a n R-linear combination of characters of RG-modules, let P be a p-subgroup of G and let e l , . . . ,en be all block idempotents e of RG with defect group P and such that x(e)/(G : P ) is invertible in R
Then we have
FG,%
rank(fp,c ) = n Proof. Let e l , ez, . . . , e n , e n + l , . . . , e m be all block idempotents of RG. Then
rank
(f:z"
=
a i=l
( f x )
(by Lemma 1.1)
Bilinear Forms on G-Algebras
=
rank i=l
(f:ga'fi)
(by Lemma 3.2)
= n(by Lemma 3.2 and Theorem 2.3(v))
as we wished t o show. H In view of the above result, it is important t o provide x for which x(e)/(G : P ) is invertible in R. This is achieved by the lemma below. The proof depends on Theorem 1.14.6 which guarantees that multiplication by e preserves ( R G ~ ~ ) ~ .
Lemma 3.4. Let Q be a Sylow p-subgroup of G and let x be the character of the RG-module ( I ~ ) If~ e. is a block idernpotent of R G with defect group P, then x(e)/(G : P ) is invertible in R
Proof. Owing to Lemma 3.2, it suffices t o show that
e g g with eg E R so that e = CsEG e g g . Because To this end, write e = CSEG B ~ $ ~ (#E0) (Lemma 1.12.13), there exists a p-regular element t of C G ( P ) such that %-I # 0. Bearing in mind that
it therefore suffices to show that
Denote by RG,I the R-linear span of the set Gpl of all pregular elements of G . Then, by Theorem 1.14.6, multiplication by e preserves ( R G , , ) ~ . Thus
which clearly implies (3). H
3 Applications to the group algebra
377
As an application, we finally prove
Corollary 3.5. (Broue' and Robinson (1986)). Let P be a p-subgroup ~ , Q is a Sylow of G , and let x be the character of the RG-module ( I ~ )where p-subgroup of G. Then the rank of the form
is equal to the number of blocks of FG with defect group P .
Proof. Apply Theorem 3.3 and Lemma 3.4. H We shall return t o the subject of counting blocks of group algebras in our future treatment of block theory.