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Cohomology of Permutation Representation in Negative Dimensions
JOURNAL OF ALGEBRA ARTICLE NO.
180, 156]174 Ž1996.
0058
Cohomology of Permutation Representation in Negative Dimensions Ixin Wen* Arizona State Uni¨ ersity, Tempe, Arizona 85287 Communicated by Walter Feit Received February 3, 1995
In 1953, Adamson discussed the cohomology group of permutation representation, Hy1 Ž G N H; G, A., where G is any group, H is a subgroup, G N H is the set of left cosets of H in G, and the action of G on G N H, known as the permutation representation, denoted by Ž G, G N H ., is transitive Žsee wAdx.. A decade later, in 1965, Snapper discussed the group Hy1 Ž G N H; G, A. again and gave the expression Hy1 Ž X ; G, A . ( a g A H Ž x . N SG N H Ž x . a s 0 4 u
= Ý SH Ž x .N H Ž x .l s i H Ž x . syi 1 sy1 y1 i is1
ž
y1 i
Ž A H Ž x .l s H Ž x . s . / , i
where X s G N H Žsee wS3x.. In this paper, we discuss the general cohomology groups H n Ž X; G, A. for all negative dimension and try to give the equivalent expression in terms of modules when Ž G, X . is not transitive. For example, we derive the formula Hy1 Ž X ; G, A . (
Ž a1 , a2 , . . . , au . N Ýujs1 SG N H Ž x . Ž ai . s 0 4 i
[
u is1
ž
Ýujs1
ij Ýtks1
SH Ž x i .N H i jk w l y 1 x Ž A H i jk .
Note. The resulting y1-cocycle group
½
Zy1 Ž X ; G, A . ( Ž a1 , a2 , . . . , a u .
u
Ý
SG N H Ž x i . Ž a i . s 0
js1
is given in wS1x. * Presently at King’s River Community College, Reedley, California. 156 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
5
.
/
157
PERMUTATION REPRESENTATION COHOMOLOGY
1. NOTATIONS A permutation representation Ž G, X . of a group G Žsee wS1x. consists of a nonempty set X on which the group G acts on the left, that is, s x g X for all s g G, x g X, and Ž1. Ž rs . x s r Ž s x . for r , s g G and x g X; Ž2. 1 x s x for x g X, where 1 is the unit element of G. We let Ž G, X . stand for a permutation representation where G and X are finite. Let Z stand for the ring of rational integers. The elements of X are denoted by x 1 , x 2 , . . . , x m and the m-fold direct sum Zw X x is the free abelian group generated by x 1 , x 2 , . . . , x m 4 . We regard Zw X x as a G-module, which we may since G acts on the Z-base X of Zw X x. G also acts on the n-fold Cartesian product X n of X Ž n ) 1. by the rule
s Ž x1 , x 2 , . . . , x n . s Ž s x1 , s x 2 , . . . , s x n . . The permutation representation Ž G, X n . and the associated G-module Zw X n x is defined as above. The standard complex of the permutation representation Ž G, X . is defined as the augmented, acyclic G-complex Ž CnŽ X; G ., n ; n g Z.: 0
Cy1Ž X ; G .
y1
Cy2 Ž X ; G .
y2
???
6
C0 Ž X ; G .
6
1
6
C 1Ž X ; G .
6
2
6
???
6 Z
m
.
6
0
66
«
0
Here, Z is regarded as a G-module with trivial action, and Ž1. CnŽ X; G . s Zw X nq 1 x for n G 0, Ž2. Cyn Ž X; G . s Zw X n x for n G 1, 1Ž Ž3. nŽ x 1 , x 2 , . . . , x nq1 . s Ý nq . jq1 Ž x 1 , . . . , ˆ x j , . . . , x nq1 ., for js1 y1 n G 1; ˆ x j indicated that x j is omitted, Ž4. 0 Ž x . s x 1 q x 2 q ??? qx m for all x g X, m Ž Ž5. yn Ž x 1 , x 2 , . . . , x n . s Ý m Ž . is1 x i , x 1 , . . . , x n y Ý is1 x 1 , x i , x 2 , . . . , Ž . x n . q ??? qŽy1. n Ý m is1 x 1 , . . . , x n , x i , for n G 1, Ž6. « Ž x . s 1 for all x g X, and Ž7. m Ž1. s x 1 q x 2 q ??? qx m .
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We refer the reader to Section 1 of wS1x for the proof that Ž CnŽ X; G ., n ; n g Z. is indeed an augmented, acyclic G-complex. We denote this complex by C?Ž X; G ., i.e., C?Ž X ; G . s Ž Cn Ž X ; G . , n ; n g Z . , where the lower dot reminds us that we are dealing with a chain complex. If A is a G-module, we denote C?Ž X ; G, A . s Ž C n Ž X ; G, A . , d n ; n g Z. , where the upper dot reminds us that we are dealing with a cochain complex, and C n Ž X; G, A. s Hom G Ž CnŽ X; G ., A. is the Z-complex, and dn s Hom G Ž nq1 , 1 A ., where 1 A indicates the identity homomorphism of the G-module A. C n Ž X; G, A. is called the nth cochain group. For all n g Z, the nth cohomology group of the Z-complex C?Ž X; G, A. is H n Ž X ; G, A . s
Ker Ž dn . Im Ž dny 1 .
s
Z n Ž X ; G, A . B n Ž X ; G, A .
.
Z n Ž X; G, A. is the nth cocycle group, and B n Ž X; G, A. is the nth coboundary group. 2. COHOMOLOGY GROUP IN DIMENSION y1 In order to study dy1 : Cy1 Ž X; G, A. ª C 0 Ž X; G, A., we must first obtain the expression for Cy1 Ž X ; G, A . s C 0 Ž X ; G, A . s Hom G Ž Z w X x , A . . Let T1 , T2 , . . . , Tu be the orbits of the permutation representation Ž G, X . Žsee wWix.. If x i s Ti for i s 1, 2, . . . , u, we refer to the set x 1 , x 2 , . . . , x u4 u as a set of representatives of the orbits Ti 4is1 . For each x g X, the subgroup of G whose elements leave x fixed is denoted by H Ž x .; i.e., H Ž x . s s g G N s x s x4 . The greatest common divisor of the indices w G: H Ž x .x is called the index of the permutation representation Ž G, X .. If x 1 , x 2 , . . . , x u4 is a set of
PERMUTATION REPRESENTATION COHOMOLOGY
159
u . representatives of Ž G, X . Žor representatives of the orbits Ti 4is1 , the study of Ž G, X . is equivalent to the simultaneous study of the classes of conjugate subgroups of G to which H Ž x 1 ., H Ž x 2 ., . . . , H Ž x u . belong. A cochain c g Hom G ŽZw X x, A. is completely determined by its values at x 1 , x 2 , . . . , x u , because, for any x g Ti , there exists s g G such that if s x i s x, then cŽ x . s cŽ s x i . s s cŽ x i .. Furthermore, for any s g H Ž x i ., if s x i s x i , then cŽ x i . s cŽ s x i . s s cŽ x i ., which implies that these values cŽ x i . cannot be chosen arbitrarily, but cŽ x i . g A H Ž x i . for i s 1, 2, . . . , u. We conclude:
PROPOSITION 2.1 ŽSee wS1x.. Cy1 Ž X ; G, A . s C 0 Ž X ; G, A . ( A H Ž x i . [ ??? [ A H Ž x u . , where [ designates the direct sum of Z-modules. If the action of G on A is tri¨ ial, then Cy1 Ž X; G, A. s C 0 Ž X; G, A. ( A [ ??? [ A, u times. We now ask: When is a y1-cochain c s Ž a1 , a2 , . . . , a u ., where a i s cŽ x i . g A H Ž x i . for i s 1, 2, . . . , u, a y1-cocycle? Knowing that for any x0 g X u
Ž dy1 c . Ž x 0 . s
Ý
u
SG N H Ž x i . c Ž x i . s
is1
Ý
SG N H Ž x i . Ž a i . ,
is1
where we use the customary symbol for the trace mapping SG N H Ž x i . : A H Ž x i . ª AG , we can easily derive that Z 0 Ž X; G, A. ( AG and B 0 Ž X; G, A. ( Ýuis1 SG N H Ž x i .Ž A H Ž x i . . Žsee wS1x.. We conclude: PROPOSITION 2.2. Zy1 Ž X; G, A. is isomorphic to the set of the u-tuples Ž a1 , a2 , . . . , a u ., where a i g A H Ž x i . for i s 1, 2, . . . , u and Ýuis1 SG N H Ž x i .Ž a i . s 0. If G acts tri¨ ially on A, then Zy1 Ž X; G, A. is isomorphic to the set of u-tuples Ž a1 , a2 , . . . , a u ., where a i g A for i s 1, 2, . . . , u, and Ýuis1w G: H Ž x i .x a i s 0. Now, we study By1 Ž X; G, A. when the permutation representation Ž G, X . is intransitive. As above, let the orbits of Ž G, X . be T1 , T2 , . . . , Tu , and T s u x 1 , x 2 , . . . , x u4 be a set of representatives of the orbits Ti 4is1 of Ž G, X .. We set H Ž xi . s s g G N s Ž xi . s xi4 ,
for x i g Ti , i s 1, 2, . . . , u,
and we let si j1 x j , si j2 x j , . . . , si jt i j x j be a set of representatives for the orbits of the permutation representation Ž H Ž x i ., Tj ., where Tj is one of the
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orbits of Ž G, X ., and x j Žin Tj . is the representation element of Tj , t i j is the number of double cosets H Ž x i .t H Ž x j . in G, where t g G, i, j s 1, 2, . . . , u, and si jk g G for k s 1, 2, . . . , t i j . PROPOSITION 2.3. The permutation representation Ž G, X 2 . has the pairs Ž x i , si jk ., for k s 1, 2, . . . , t i j , i, j s 1, 2, . . . , u, for a set of representati¨ es of its orbits, where X 2 is the Cartesian product of X with itself. Proof. We need to prove that:
s Ž x i , si jk x j . / Ž x i9 , si9 j9k 9 x j9 . for all s g G. We have three cases: Case 1. If i / i9, clearly s Ž x i . / x i9 for all s g G because x i and x i9 are not in the same orbit of Ž G, X .. Case 2. If i s i9 and j / j9, then we have
ssi jk x j / si j9k 9 x j9 , for all s g G, since x j and x j9 , j / j9, are in different orbits of Ž G, X .. Case 3. If i s i9, j s j9, but k / k9, we assume
s Ž x i , si jk x j . s Ž x i , si jk 9 x j . for some s g G. Then we know that s Ž x i . s x i , which means s must be in H Ž x i . and ssi jk x j s si jk 9 x j , which is impossible because si jk x j and si jk 9 x j , k / k9, are in different orbits of Ž H Ž x i ., Tj .. Now, consider an arbitrary pair Ž s x i , t x j . of X 2 , where s , t g G; then we have Ž x i , sy1t x j . s sy1 Ž s x i , t x j .. Since sy1t x j g Tj , there exists a r g H Ž x i . such that rsy1t x j s si jk x j for some k Ž1 F k F t i j ., then Ž x i , si jk x j . s rsy1 Ž s x i , t x j ., and the proposition is proved. According to Propositions 2.1 and 2.2, a cochain, c g Cy2 Ž X; G, A. s Hom G ŽZw X 2 x, A., is completely determined by its value at the pairs Ž x i , si jk x j .. These values cŽ x i , si jk x j . cannot be chosen arbitrarily, but cŽ x i , si jk x j . g A H i jk , where Hi jk s H Ž x i . l si jk H Ž x j . sy1 i jk . Since the cochain c is a G-homomorphism and if s Ž x i , si jk x j . s Ž x i , si jk x j . for some s g G, we must have c Ž s Ž x i , si jk x j . . s s c Ž x i , si jk x j . s c Ž x i , si jk x j . ,
PERMUTATION REPRESENTATION COHOMOLOGY
161
it implies that s Ž x i . s x i , ssi jk x j s si jk x j ; in other words, s g H Ž x i . and Ž . Ž . y1 s g si jk H Ž x j . sy1 i jk , i.e., s g H x i l si jk H x j si jk . We conclude ti j PROPOSITION 2.4. Cy2 Ž X; G, A. ( [i,u js1 [ks1 A H i jk , where A H i jk s a g A N s a s a for all s g Hi jk 4 .
From Proposition 2.4 we know that a cochain c g Cy2 Ž X; G, A. is completely determined by its values at the pairs Ž x i , si jk x j .; in particular, its values at the pairs Ž si jk x j , x i .. We like to know the relation between these pairs. Let us begin with the pair Ž si jk x j , x i ., we have
Ž si jk x j , x i . s si jk Ž x j , sy1 i jk x i . , there exists r g H Ž x j . such that
ry1sy1 i jk x i s sji k 9 x i
Ž 1.1.
for some k9 belonging to 1, 2, . . . , t ji 4 , and, obviously, k9 depends on ry1sy1 i jk , where si jk x i belongs to the set of representatives for the orbits of Ž H Ž x i ., Tj ., and sji k 9 x i belongs to the set of representatives for the orbits of Ž H Ž x j ., Ti .; therefore, we conclude, y1 y1 Ž si jk x j , x i . s si jk Ž x j , sy1 i jk x i . s si jk Ž r x j , rr si jk x i . y1 s si jk r Ž x j , ry1sy1 i jk x i . s sji k 9 Ž x j , sji k 9 x i . .
For convenience, we define the following mapping:
l : Ž x i , si jk x j . 4 ª Ž si jk x j , x i . 4 Ž . such that lŽ x i , si jk x j . s Ž si jk x j , x i . s sy1 ji k 9 x j , sji k 9 x i . If we set c Ž x i , si jk x j . s a i jk g A H i jk , c Ž x j , sji k 9 x i . s a ji k 9 g A H ji k 9 ; then
l a i jk s l c Ž x i , si jk x j . s c l Ž x i , si jk x j . s c Ž si jk x j , x j . s c sy1 ji k 9 Ž x j , sji k 9 x i . y1 s sy1 ji k 9 c Ž x j , sji k 9 x i . s sji k 9 a ji k 9 .
Ž 1.2.
162
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LEMMA 2.1.
l a i jk g A H i jk if and only if a ji k 9 g A H ji k 9.
Proof. a ji k 9 g A H ji k 9 m Ž H Ž x j . l sji k 9 H Ž x i . sy1 ji k 9 . a ji k 9 s a ji k 9 m Ž H Ž x j . sji k 9 l sji k 9 H Ž x i . . sy1 ji k 9 a ji k 9 s a ji k 9 y1 y1 m sy1 ji k 9 Ž H Ž x j . sji k 9 l sji k 9 H Ž x i . . sji k 9 a ji k 9 s sji k 9 a ji k 9 y1 y1 m Ž sy1 ji k 9 H Ž x j . sji k 9 l H Ž x i . .Ž sji k 9 a ji k 9 . s sji k 9 a ji k 9 ,
from Ž1.1. we have
ry1sy1 i jk x i s sji k 9 x i ,
r g HŽ xj.
i.e.,
ry1sy1 i jk s sji k 9 . y1 Ž ..Ž sy1 . Therefore, Ž si jk r H Ž x j . ry1sy1 i jk l H x i i jk 9 a ji k 9 s sji k 9 a ji k 9 , and from Ž1.2. we have
sy1 ji k 9 a ji k 9 s l a i jk , hence
Ž si jk H Ž x j . sy1 i jk l H Ž x i . . Ž l a i jk . s l a i jk , since r g H Ž x j ., r H Ž x j . ry1 s H Ž x j ., and Hi jk s H Ž x i . l si jk H Ž x j . sy1 i jk ; therefore, Hi jk Ž l a i jk . s l a i jk m l a i jk g A H i jk . Now we define the mapping ti j
u
Li:
[ [A js1 ks1
H i jk
ª AH Ž x i. ,
for i s 1, 2, . . . , u, by L i Ž a i jk N k s 1, 2, . . . , t i j ; j s 1, 2, . . . , u . u
s
ti j
Ý Ý
S H Ž x i .N H i jk Ž l a i jk y a i jk .
js1 ks1 u
s
ti j
Ý Ý js1 ks1
SH Ž x i .N H i jk w l y 1 x Ž a i jk . ,
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PERMUTATION REPRESENTATION COHOMOLOGY
where the mapping l is given in Ž1.2., 1 is the identity mapping, and SH Ž x i .N H i jk is the trace mapping from A H i jk to A H Ž x i ., and Ž a i jk N k s u ti j 1, 2, . . . , t i j ; j s 1, 2, . . . , u. belongs to [js1 [ks1 A H i jk . In short, we write ti j
u
Li s
SH Ž x i .N H i jk w l y 1 x
Ý Ý js1 ks1
for i s 1, 2, . . . , u. According to Lemma 2.1, we know that l a i jk g A H i jk , therefore L i is well-defined for i s 1, 2, . . . , u. Consider the following diagrams S 1 and S 2 : 6
Cy1 Ž X ; G, A . 6
S2
py2
py1
ti j
u
C 0 Ž X ; G, A . 6
S1
p0
u
dy2
A [ is1
u
H Ž xi.
6
H i jk
6
[ [A i , js1 ks1
d y1
6
dy2
Cy2 Ž X ; G, A . 6
dy1
A [ is1
H Ž xi.
where the mapping dy2 is defined by u
dy2 s
L; [ is1 i
this means that for any ti j
u
Ž ai jk N k s 1, 2, . . . , t i j ; i , j s 1, 2, . . . , u . g
[ [A
H i jk
,
i , js1 ks1
dy2 Ž a i jk N k s 1, 2, . . . , t i j ; i , j s 1, 2, . . . , u . u
s s
L Ža [ is1 i
ž
t1 j
u
Ý Ý
i jk
N k s 1, 2, . . . , t i j ; j s 1, 2, . . . , u .
S H Ž x 1 .N H 1 jk w l y 1 x Ž a1 jk . ,
js1 ks1
t2 j
u
Ý Ý
= w l y 1 x Ž a2 jk . , . . . ,
u
tu j
Ý Ý
SH Ž x u .N H u jk w l y 1 x Ž a u jk .
js1 ks1 u
s
u
ti j
[ Ý Ý is1
ž
S H Ž x 2 .N H 2 jk
js1 ks1
js1 ks1
S H Ž x i .N H i jk w l y 1 x Ž a i jk . .
/
/ Ž 1.3.
164
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The mapping py2 is defined by Py2 Ž a i jk N k s 1, 2, . . . , t i j ; i , j s 1, 2, . . . , u . Ž x i , si jk x j . s a i jk . The mappings dy2 and dy1 are defined as before. PROPOSITION 2.5. The diagram S 1 is commutati¨ e; in other words, dy1 ( py1 s po ( dy1 , where py1 , po are isomorphisms Ž see Proposition 2.1, actually p1 s p 2 . and defined by pj Ž Ž a1 , a2 , . . . , a u . . Ž x i . s a i ; j s y1, 0 and i s 1, 2, . . . , u. dy1 s D uŽÝuis1 SG N H Ž x i . ., where, D u is a diagonal mapping, D uŽ g . s Ž g, g, . . . , g ., u times. u
dy1 Ž Ž a1 , a2 , . . . , a u . . s D u
žÝ
SG N H Ž x i . Ž a i .
is1
u
s
žÝ
/ u
SG N H Ž x i . Ž a i . ,
is1
Ý
SG N H Ž x i . Ž a i . , . . . ,
is1 u
= Ý SG N H Ž x i . Ž a i . .
/
is1
Proof. For any Ž a1 , a2 , . . . , a u . g A H Ž x 1 . [ ??? [ A H Ž x u .
Ž dy1 ( py1 . Ž Ž a1 , a2 , . . . , au . . Ž x i . s py1 Ž Ž a1 , a2 , . . . , a u . . ( 0 Ž x i . s py1 Ž Ž a1 , a2 , . . . , a u . . Ž x 1 q x 2 q ??? qx m . u
s py1 Ž Ž a1 , a2 , . . . , a u . .
ž
Ý is1
SG N H Ž x i . Ž x i .
/
u
s
Ý
SG N H Ž x i . Ž py1 Ž Ž a1 , a2 , . . . , a u . . Ž x i . .
is1 u
s
Ý is1
SG N H Ž x i . Ž a i . .
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PERMUTATION REPRESENTATION COHOMOLOGY
On the other hand,
Ž p 0 ( dy1 . Ž Ž a1 , a2 , . . . , au . . Ž x i . s p 0 Ž dy1 Ž Ž a1 , a2 , . . . , a u . . . Ž x i . u
s p0
ž
Ý
u
SG N H Ž x i . Ž a i . , . . . ,
is1
/
SG N H Ž x i . Ž a i . Ž x i .
Ý is1
u
s
Ý
SG N H Ž x i . Ž a i . .
is1
Therefore, dy1 ( py1 s po ( dy1. THEOREM 2.1. The diagram S 2 is commutati¨ e. Proof. For any u t [ks1 Ž ai jk N k s 1, 2, . . . , t i j ; i , j s 1, 2, . . . , u . g [js1 ij
A H i jk ,
we set f Ž x i . s Ž dy2 ( py2 . Ž a i jk N k s 1, 2, . . . , t i j ; i , j s 1, 2, . . . , u . Ž x i . s py2 Ž a i jk N k s 1, 2, . . . , t i j ; i , j s 1, 2, . . . , u . ( y1 Ž x i . s py2 Ž a i jk N k s 1, 2, . . . , t i j ; i , j s 1, 2, . . . , u . =
žÝŽ x lgX
xl , xi . y
Ý Ž xi , xl .
x lgX
/
s py2 Ž a i jk N k s 1, 2, . . . , t i j ; i , j s 1, 2, . . . , u . =
ž
u
ti j
Ý Ý
ti j
u
SH Ž x i .N H i jkŽ si jk x j , x i . y
js1 ks1
Ý Ý
SH Ž x i .N H i jkŽ x i , si jk x j .
js1 ks1
s py2 Ž a i jk N k s 1, 2, . . . , t i j ; i , j s 1, 2, . . . , u . = u
s
ž
u
ti j
Ý Ý
SH Ž x i .N H i jk Ž lŽ x i , si jk x j . y Ž x i , si jk x j . .
js1 ks1 ti j
Ý Ý js1 ks1
u
SH Ž x i .N H i jk Ž l a i jk y a i jk . s
/
ti j
Ý Ý
SH Ž x i .N H i jk
js1 ks1
= w l y 1 x Ž a i jk . ,
/
166
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for i s 1, 2, . . . , u; on the other hand, we put g Ž x i . s Ž py1 ( dy2 . Ž a i jk N k s 1, 2, . . . , t i j ; i , j s 1, 2, . . . , u . Ž x i . , from Ž1.3. and the definition of py1 , u
g Ž x i . s py1 u
s
ž[ž is1 ti j
ti j
u
Ý Ý
SH Ž x i .N H i jk w l y 1 x Ž a i jk .
js1 ks1
//
Ž xi .
SH Ž x i .N H i jk w l y 1 x Ž a i jk . .
Ý Ý js1 ks1
Hence, f Ž x i . s g Ž x i ., for all i s 1, 2, . . . , u, which implies that f s g; i.e., py1 ( dy2 s dy2 ( py2 . THEOREM 2.2. By1 Ž X ; G, A . ( Im dy2 s u
s
u
Im L [ is1
u
t it
[ Ý Ý is1
ž
i
SH Ž x i .N H i jk w l y 1 x Ž A H i jk . .
js1 ks1
/
Proof. Since S 2 is commutative, py2 and py1 are isomorphisms, and the theorem follows. COROLLARY 2.1.
dy1 ( dy2 s 0.
Proof. Since dy1 ( dy2 s 0, and S1 and S 2 are commutative,
dy1 ( Ž dy2 ( py2 . s dy1 ( Ž py1 ( dy2 . , thus
dy1 ( dy2 ( py2 s Ž p 0 ( dy1 . ( dy2 , and dy1 ( dy2 s 0 since py2 and p 0 are isomorphisms. Remark 2.1. The ‘‘trace mappings’’ dy1 and dy2 Žbecause dy1 and dy2 consist of trace mappings. depend on the choice of representatives
167
PERMUTATION REPRESENTATION COHOMOLOGY
x 1 , x 2 , . . . , x u of Ž G, X . and the pairs Ž x i , si jk x j . N k s 1, 2, . . . , t i j ; i, j s 1, 2, . . . , u4 . We may henceforth say that dy1 and dy2 are determined up to isomorphisms. The computation of By1 Ž X; G, A., in the case that Ž G, X . is transitive, is given on page 69 of wAdx and in Section 11 of wS3x. The following theorem is an immediate consequence of Proposition 2.2, Theorem 2.2, and the definition of dy1 Žsee Proposition 2.5.. THEOREM 2.3. Hy1 Ž X ; G, A . (
Ž a1 , a2 , . . . , au . N Ýujs1 SG N H Ž x . Ž ai . s 0 4 i
[
u is1
ž
Ýujs1
ij Ýtks1
SH Ž x i .N H i jk w l y 1 x Ž A H i jk .
.
/
Ž For Ž G, X . transiti¨ e, see Theorem 4.3 in w Ad x..
3. COHOMOLOGY OF PERMUTATION REPRESENTATION IN NEGATIVE DIMENSIONS In the previous section, we discussed cohomology in dimension y1; now we use the induction method to study more general situations. For convenience, we adopt the following notations: I1 is a set of representatives of orbits for permutation representation Ž G, X ., I1 : X. Ž x i1 , x i2 . N x i1 g I1 , x i2 g I2 4 is a set of representatives of orbits for Ž G, X 2 ., where I1 : I2 : X; from Proposition 2.3, the pairs Ž x i , si jk x j ., for si jk g G, k s 1, 2, . . . , t i j , i, j s 1, 2, . . . , u, form a representative set for the orbits of Ž G, X 2 .; x i , x j g I1. We put I2 s si jk x j N k s 1, 2, . . . , t i j ; i , j s 1, 2, . . . , u4 and, without loss of generality, we can always choose si j1 x j s x j , i.e., si j1 s 1 g G. We assume that if Ž x i1 , x i2 , . . . , x i q . N x i1 g I1 , x i2 g I2 , . . . , x i q g Iq , I1 : I2 : ??? : Iq : X 4 is a representative set of orbits for Ž G, X q ., then there exists Iqq 1 such that Iq : Iqq1 : X, and Ž x i1 , x i2 , . . . , x i q , x i qq1 . N x i j g I j , I j : I jq1 , j s 1, 2, . . . , q, and x i qq1 g Iqq1 4 is a representative set of orbits for Ž G, X qq 1 .. The construction of Iqq 1 is as follows. Let H Ž x i1 , x i2 , . . . , x i q . s s g G N s Ž x i1 , x i2 , . . . , x i q . s Ž x i1 , x i2 , . . . , x i q .4 . Clearly, H Ž x i1 , x i2 , . . . , x i q . s H Ž x i1 . l H Ž x i2 . l ??? l H Ž x i q .. Let the set x i1, i2, . . . , i q, k N k s 1, 2, . . . , t i1, i2, . . . , i q,4 : X be a repre-
168
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sentative set of orbits for Ž H Ž x i1 , x i2 , . . . , x i q ., X . where t i1, i2, . . . , i q is the number of orbits of Ž H Ž x i1 , x i2 , . . . , x i q ., X .. Since k G 1, we can choose x i1, i2, . . . , i q, 1 s x i q for some x i1 , x i2 , . . . , and x i q . Hence, Iqq1 s Dqjs1 Dx i j g I j x i1, i2, . . . , i q, k N k s 1, 2, . . . , t i1, i2, . . . , i q,4 , and by the construction, Iq : Iqq1 : X. LEMMA 3.1. Ž x i1 , x i2 , . . . , x i q , x i qq1 . N x i j g I j , j s 1, 2, . . . , q q 14 is a representati¨ e set of orbits for Ž G, X qq 1 .. Proof. We need to prove that for any s g G,
s Ž x i1 , x i2 , . . . , x i q , x i qq1 . / Ž x i19 , x i2 9 , . . . , x i q9 , x i qq19 . , where x i j , x i j9 g I j , j s 1, 2, . . . , q q 1. If Ž x i1 , x i2 , . . . , x i q . / Ž x i19 , x i2 9 , . . . , x i q9 ., then for all s g G, s Ž x i1 , x i2 , . . . , x i q , x i qq1 . / Ž x i19 , x i2 9 , . . . , x i q9 , x i qq19 . because Ž x i1 , x i2 , . . . , x i q . and Ž x i19 , x i2 9 , . . . , x i q9 . are different orbits of Ž G, X q .. If Ž x i1 , x i2 , . . . , x i q . s Ž x i19 , x i2 9 , . . . , x i q9 ., and x i qq1 / x i qq19 , suppose that there exists a s g G such that
s Ž x i1 , x i2 , . . . , x i q , x i qq1 . s Ž x i1 , x i2 , . . . , x i q , x i qq19 . , which implies that s g H Ž x i1 , x i2 , . . . , x i q . and s x i qq1 s x i qq19 , x i qq1 , x i qq19 g Iqq1 , which is impossible. For any Ž x 1 , x 2 , . . . , x q , x qq1 ., x j g X, j s 1, 2, . . . , q q 1, because Ž x 1 , x 2 , . . . , x q . g X q there exists s g G such that
s Ž x 1 , x 2 , . . . , x q , x qq1 . s Ž x i1 , x i2 , . . . , x i q , s x qq1 . , where x i j g I j , j s 1, 2, . . . , q. Furthermore, there exists H Ž x i1 , x i2 , . . . , x i q . such that rs x qq1 s x i qq1 g Iqq1. Putting the above together, we have
rs Ž x 1 , x 2 , . . . , x q , x qq1 . s r Ž x i1 , x i2 , . . . , x i q , s x qq1 . s Ž x i1 , x i2 , . . . , x i q , rs x qq1 . s Ž x i1 , x i2 , . . . , x i q , x i qq1 . . Therefore, the lemma is proved.
rg
PERMUTATION REPRESENTATION COHOMOLOGY
169
It follows from the above lemma that we have PROPOSITION 3.1. q
Cyq Ž X ; G, A . (
[ [A js1 x gI ij
H Ž x i1 , x i2 , . . . , x i q .
,
j
where A H Ž x i1 , x i2 , . . . , x i q . s a g A N s a s a for all s g H Ž x i1 , x i2 , . . . , x i q .4 . Next we define a family of mappings qq 1 l k N k s 0, 1, . . . , q4 on X qq 1. qq 1 l k
Ž x i1 , x i2 , . . . , x i q , x i qq1 . s Ž x i1 , x i2 , . . . , x i k , x i qq1 , x i kq1 , . . . , x i q . ,
then there exists s g H Ž x i1 , x i2 , . . . , x i k . such that
Ž x i1 , x i2 , . . . , x i k , x i qq1 , x i kq1 , . . . , x i q . s s Ž x i1 , x i2 , . . . , x i k , x i kq10 , . . . , x i q0 , x i qq10 . , where x i j g I j , j s 1, 2, . . . , q, q q 1, and x i j0 g I j , j s k q 1, . . . , q q 1; i.e., qq 1 l k
Ž x i1 , x i2 , . . . , x i q , x i qq1 .
s s Ž x i1 , x i2 , . . . , x i k , x i kq10 , . . . , x i q0 , x i qq10 . . For any c s CyŽ qq1. Ž X; G, A., obviously, qq 1 l k
Ž c Ž x i1 , x i2 , . . . , x i q , x i qq1 . .
s s c Ž x i1 , x i2 , . . . , x i k , x i kq10 , . . . , x i q0 , x i qq10 . . Therefore,
qq 1 l k
is uniquely determined by the value of c on the set
Ž x i1 , x i2 , . . . , x i q , x i qq1 . N x i j g Ij , j s 1, 2, . . . , q q 1 4 . For all x i qq1 g Iqq1 , we assume that c Ž x i1 , x i2 , . . . , x i q , x i qq1 . s a i1, i2, . . . , i qq1 and c Ž x i1 , x i2 , . . . , x i k , x i kq10 , . . . , x i q0 , x i qq10 . s a i1, i2, . . . , i k , i kq10 , . . . , i qq10 , then qq 1 l k
Ž ai1, i2, . . . , i qq1 . s s ai1, i2, . . . , i k , i kq10 , . . . , i qq10 ,
for some s g H Ž x i1 , x i2 , . . . , x i k ..
170
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LEMMA 3.2.
qq 1 l k
Ž a i1, i2, . . . , i qq1 . g A H Ž x i1 , x i2 , . . . , x i qq 1 . if and only if
a i1, i2, . . . , i k , i kq10 , . . . , i qq10 g A H Ž x i1 , x i2 , . . . , x i k , x i kq 10 , . . . , x i q 0 , x i qq 10 . , k s 0, 1, . . . , q. Proof. Because a i1, i2, . . . , i k , i kq10 , . . . , i qq10 g A H Ž x i1 , x i2 , . . . , x i k , x i kq 10 , . . . , x i q 0 , x i qq 10 . m H Ž x i1 , x i2 , . . . , x i k , x i kq10 , . . . , x i q0 , x i qq10 . = a i1, i2, . . . , i k , i kq10 , . . . , i qq10 s a i1, i2, . . . , i k , i kq10 , . . . , i qq10 m s H Ž x i1 , x i2 , . . . , x i k , x i kq10 , . . . , x i q0 , x i qq10 . = sy1 Ž s a i1, i2, . . . , i k , i kq10 , . . . , i qq10 . s s a i1, i2, . . . , i k , i kq10 , . . . , i qq10 m s H Ž x i1 , x i2 , . . . , x i k , x i kq10 , . . . , x i q0 , x i qq10 . = sy1 Ž qq 1 l k Ž a i1, i2, . . . , i qq1 . . sqq 1 l k Ž a i1, i2, . . . , i qq1 . . Now we only need to prove that
s H Ž x i1 , x i2 , . . . , x i k , x i kq10 , . . . , x i q0 , x i qq10 . sy1 s H Ž x i1 , x i2 , . . . , x i q , x i qq1 . . Indeed,
s H Ž x i1 , x i2 , . . . , x i k , x i kq10 , . . . , x i q0 , x i qq10 . sy1 s s H Ž x i1 . sy1 l s H Ž x i2 . sy1 l ??? l s H Ž x i k . sy1 l s H Ž x i kq10 . sy1 l ??? l s H Ž x i qq10 . sy1 s H Ž x i1 . l H Ž x i2 . l ??? l H Ž x i k . l H Ž x i qq1 . l H Ž x i kq1 . l ??? l H Ž x i q . s H Ž x i1 , x i2 , . . . , x i k , x i kq1 , . . . , x i q , x i qq1 . because s x i kq10 s x i qq1 , s x i kq2 0 s x i kq1 , . . . , s x i qq10 s x i q . It implies that qq 1 l k Ž a i1, i2, . . . , i qq1 . g A H Ž x i1 , x i2 , . . . , x i qq 1 ..
171
PERMUTATION REPRESENTATION COHOMOLOGY
We now define the following mapping: L i1, i2, . . . , i q :
[
x iqq1 gI qq1
A H Ž x i1 , x i2 , . . . , x i q , x i qq 1 . ª A H Ž x i1 , . . . , x i q .
by L i1, i2, . . . , i q Ž a i1 , i2, . . . , i qq1 N x i qq1 g Iqq1 . s
Ý
SH Ž x i1 , x i2 , . . . , x i q .N H Ž x i1 , x i2 , . . . , x i q , x i qq 1 .
qq 1 l 0
yqq1 l1 q ??? q Ž y1 .
x i qq1 gI qq1
=
q
qq1 l q
Ž ai1, i2, . . . , i qq1 . .
In short, we write L i1 , i2, . . . , i q s
Ý
SH Ž x i1 , x i2 , . . . , x i q .N H Ž x i1 , x i2 , . . . , x i q , x i qq 1 .
qq 1 l 0
yqq1 l1 q ??? q Ž y1 .
x i qq1 gI qq1
=
q
qq1 l q
,
for x i j g I j , j s 1, 2, . . . , q. According to Lemma 3.2, we know that L i1, i2, . . . , i q is well defined. THEOREM 3.1. The following diagram is commutati¨ e: dyŽ qq1.
Cyq Ž X ; G, A . 6
6
CyŽ qq1. Ž X ; G, A . 6 pyŽ qq1.
pyq
qq1
ij
H Ž x 1 , x 2 , . . . , x i q , x i qq 1 .
dyŽ qq1.
j
q where dyŽ qq1. s [js1
[x
i j g Ij
[ [A js1 x gI
6
[ [A js1 x gI
q
ij
H Ž x1 , x 2 , . . . , x i q .
j
L i1, i2, . . . , i q .
Proof. For any a i1, i2, . . . , i qq1 g A H Ž x i1 , x i2 , . . . , x i qq 1 ., we compute
Ž Ž dyŽ qq1. ( pyŽ qq1. .Ž ai1, i2, . . . , i qq1 N x i j g Ij , j s 1, 2, . . . , q q 1. . = Ž x i1 , x i2 , . . . , x i q . s Ž pyŽ qq1. Ž a i1, i2, . . . , i qq1 N x i j g I j , j s 1, 2, . . . , q q 1 . . =( yq Ž x i1 , x i2 , . . . , x i q . s Ž pyŽ qq1. Ž a i1, i2, . . . , i qq1 N x i j g I j , j s 1, 2, . . . , q q 1 . .
,
172
IXIN WEN
=
ž Ý Ž x, x , x i1
i2 , . . . ,
xiq . y
Ý Ž x i1 , x, x i2 , . . . , x i q .
xgX
xgX q
q ??? q Ž y1 . Ž x i1 , x i2 , . . . , x i q , x .
/
s Ž pyŽ qq1. Ž a i1, i2, . . . , i qq1 N x i j g I j , j s 1, 2, . . . , q, q q 1 . .
ž
Ý
x i qq1 gI qq1
y
Ž SH Ž x
i1 ,
x i2 , . . . , x i q .N H Ž x i1 , x i2 , . . . , x i qq 1 .
Ž SH Ž x
Ý
x iqq1 gI qq1
q ??? q Ž y1 .
x i2 , . . . , x i q .N H Ž x i1 , x i2 , . . . , x i qq 1 .
i1 ,
q
Ž x i qq1 , x i1 , x i2 , . . . , x i q . .
Ý
x iqq1 gI qq1
Ž SH Ž x
i1 ,
Ž x i1 , x i qq1 x i2 , . . . , x i q . .
x i2 , . . . , x i q .N H Ž x i1 , x i2 , . . . , x i qq 1 .
= Ž x i1 , x i2 , . . . , x i q , x i qq1 . .
/
s Ž pyŽ qq1. Ž a i1 , i2, . . . , i qq1 N x i j g I j , j s 1, 2, . . . , q, q q 1 . .
ž
Ý
x iqq1 gI qq1
žS
H Ž x i1 , x i2 , . . . , x i q .N H Ž x i1 , x i2 , . . . , x i qq 1 .
=
qq 1 l 0
yqq1 l1 q ??? q Ž y1 .
= Ž x i1 , x i2 , . . . , x i q , x i qq1 . s
Ý
x iqq1 gI qq1
žS
q
qq1 l q
//
H Ž x i1 , x i2 , . . . , x i q .N H Ž x i1 , x i2 , . . . , x i qq 1 .
=
qq 1 l 0
yqq1 l1 q ??? q Ž y1 .
= Ž a i1, i2, . . . , i qq1 .
q
qq1 l q
/
On the other hand, we have
Ž Ž pyq ( dyŽ qq1. .Ž ai1, i2, . . . , i qq1 N x i j g Ij , j s 1, 2, . . . , q q 1. . = Ž x i1 , x i2 , . . . , x i q . q
s pyq
ž[ Ý js1
x i jgI j
L i1, i2, . . . , i q Ž a i1, i2, . . . , i qq1 N x i qq1 g Iqq1 .
= Ž x i1 , x i2 , . . . , x i q .
/
173
PERMUTATION REPRESENTATION COHOMOLOGY
s L i1, i2, . . . , i q Ž a i1, i2, . . . , i qq1 N x i qq1 g Iqq1 . s
Ý
x iqq1 gI qq1
žS =
H Ž x i1 , x i2 , . . . , x i q .N H Ž x i1 , x i2 , . . . , x i qq 1 .
qq 1 l 0
yqq1 l1 q ??? q Ž y1 .
q
qq1 l q
Ž ai1, i2, . . . , i qq1 . .
/
Therefore, we have proved the theorem. COROLLARY.
dyq ( dyŽ qq1. s 0.
Proof. It follows from dyq ( dyŽ qq1. s 0. THEOREM 3.2. Zyq Ž X ; G, A . (
½Ž
a i1 , i2, . . . , i q N x i j g I j , j s 1, 2, . . . , q . N a i1, i2, . . . , i q
g A H Ž x i1 , x i2 , . . . , x i q . ,
Ý
x iqgI q
S H Ž x i1 , x i2 , . . . , x i qy 1 .N H Ž x i1 , x i2 , . . . , x i q .
= q l0 yq l1 q ??? q Ž y1 .
qy 1
q l qy1
Ž ai1, i2, . . . , i q . s 0
5
for x i j g I j , j s 1, 2, . . . , q y 1 . q
Byq Ž X ; G, A . (
[ [ js1 x gI ij
S H Ž x i1 , x i2 , . . . , x i q .N H Ž x i1 , x i2 , . . . , x i qq 1 .
Ý
j
x i qq1 gI qq1
=
qq 1 l 0
yqq1 l1 q ??? q Ž y1 .
q
qq1 l q
= Ž A H Ž x i1 , x i2 , . . . , x i1 , x i qq 1 . . . In short, Hyq Ž X ; G, A . ( Ker dyq rIm dyŽ qq1. qy1
s
[ [ Ker Ž L js1 x gI ij
q
i1, i2, . . . , i qy1
.
j
[ [ Im Ž L js1 x gI ij
i1, i2, . . . , i q
..
j
REFERENCES wAdx wBax
I. T. Adamson, Cohomology theory for non-normal subgroups and non-normal fields, Proc. Glasgow Math. Assoc. 2 Ž1954., 66]76. A. Babakhanian, ‘‘Cohomological Methods in Group Theory,’’ Dekker, New York, 1972.
174 wBex
IXIN WEN
D. J. Benson, ‘‘Representations and Cohomology,’’ Vols. I, II, Cambridge Univ. Press, Cambridge, 1991. wBwx N. L. Biggs and A. T. White, ‘‘Permutation Groups and Combinatorial Structures,’’ Cambridge Univ. Press, Cambridge, 1979. wBrx K. S. Brown, ‘‘Cohomology of Groups,’’ Springer-Verlag, New York, 1982. wEvx L. Evens, ‘‘The Cohomology of Groups,’’ Oxford Univ. Press, Oxford, 1991. wHSx P. J. Hilton and U. Stammback, ‘‘A Course in Homological Algebra,’’ Springer-Verlag, New York, 1971. wMax S. MacLane, ‘‘Homology,’’ Springer-Verlag, Berlin, 1963. wRox J. J. Rotman, Homology groups of Steiner systems, J. Algebra 92 Ž1985., 128]149. wS1x E. Snapper, Cohomology of permutation representations. I. Spectral sequences, J. Math. Mech. 13 Ž1964., 133]161. wS2x E. Snapper, Cohomology of permutation representations. II. Cup product, J. Math. Mech. 13 Ž1964., 1047]1064. wS3x E. Snapper, Inflation and deflation for all dimensions, Pacific J. Math. 15 Ž1965., 1061]1081. wS4x E. Snapper, Duality in the cohomology ring of transitive permutation representations, J. Math. Mech. 14 Ž1965., 323]336. wS5x E. Snapper, Spectral sequence and Frobenius groups, Trans. Amer. Math. Soc. 114 Ž1965., 113]146. wSux M. Suzuki, ‘‘Group Theory,’’ Vols. I, II, Springer-Verlag, New York, 1982. wTh1x C. B. Thomas, ‘‘Characteristic Classes and the Cohomology of Finite Groups,’’ Cambridge Univ. Press, Cambridge, 1986. wWx I. Wen, ‘‘Cohomology of Permutation Representation in Combinatorial Designs,’’ dissertation, Arizona State University, May 1993. wWix H. Wielandt, ‘‘Finite Permutation Groups,’’ Academic Press, New York, 1964.
180, 156]174 Ž1996.
0058
Cohomology of Permutation Representation in Negative Dimensions Ixin Wen* Arizona State Uni¨ ersity, Tempe, Arizona 85287 Communicated by Walter Feit Received February 3, 1995
In 1953, Adamson discussed the cohomology group of permutation representation, Hy1 Ž G N H; G, A., where G is any group, H is a subgroup, G N H is the set of left cosets of H in G, and the action of G on G N H, known as the permutation representation, denoted by Ž G, G N H ., is transitive Žsee wAdx.. A decade later, in 1965, Snapper discussed the group Hy1 Ž G N H; G, A. again and gave the expression Hy1 Ž X ; G, A . ( a g A H Ž x . N SG N H Ž x . a s 0 4 u
= Ý SH Ž x .N H Ž x .l s i H Ž x . syi 1 sy1 y1 i is1
ž
y1 i
Ž A H Ž x .l s H Ž x . s . / , i
where X s G N H Žsee wS3x.. In this paper, we discuss the general cohomology groups H n Ž X; G, A. for all negative dimension and try to give the equivalent expression in terms of modules when Ž G, X . is not transitive. For example, we derive the formula Hy1 Ž X ; G, A . (
Ž a1 , a2 , . . . , au . N Ýujs1 SG N H Ž x . Ž ai . s 0 4 i
[
u is1
ž
Ýujs1
ij Ýtks1
SH Ž x i .N H i jk w l y 1 x Ž A H i jk .
Note. The resulting y1-cocycle group
½
Zy1 Ž X ; G, A . ( Ž a1 , a2 , . . . , a u .
u
Ý
SG N H Ž x i . Ž a i . s 0
js1
is given in wS1x. * Presently at King’s River Community College, Reedley, California. 156 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
5
.
/
157
PERMUTATION REPRESENTATION COHOMOLOGY
1. NOTATIONS A permutation representation Ž G, X . of a group G Žsee wS1x. consists of a nonempty set X on which the group G acts on the left, that is, s x g X for all s g G, x g X, and Ž1. Ž rs . x s r Ž s x . for r , s g G and x g X; Ž2. 1 x s x for x g X, where 1 is the unit element of G. We let Ž G, X . stand for a permutation representation where G and X are finite. Let Z stand for the ring of rational integers. The elements of X are denoted by x 1 , x 2 , . . . , x m and the m-fold direct sum Zw X x is the free abelian group generated by x 1 , x 2 , . . . , x m 4 . We regard Zw X x as a G-module, which we may since G acts on the Z-base X of Zw X x. G also acts on the n-fold Cartesian product X n of X Ž n ) 1. by the rule
s Ž x1 , x 2 , . . . , x n . s Ž s x1 , s x 2 , . . . , s x n . . The permutation representation Ž G, X n . and the associated G-module Zw X n x is defined as above. The standard complex of the permutation representation Ž G, X . is defined as the augmented, acyclic G-complex Ž CnŽ X; G ., n ; n g Z.: 0
Cy1Ž X ; G .
y1
Cy2 Ž X ; G .
y2
???
6
C0 Ž X ; G .
6
1
6
C 1Ž X ; G .
6
2
6
???
6 Z
m
.
6
0
66
«
0
Here, Z is regarded as a G-module with trivial action, and Ž1. CnŽ X; G . s Zw X nq 1 x for n G 0, Ž2. Cyn Ž X; G . s Zw X n x for n G 1, 1Ž Ž3. nŽ x 1 , x 2 , . . . , x nq1 . s Ý nq . jq1 Ž x 1 , . . . , ˆ x j , . . . , x nq1 ., for js1 y1 n G 1; ˆ x j indicated that x j is omitted, Ž4. 0 Ž x . s x 1 q x 2 q ??? qx m for all x g X, m Ž Ž5. yn Ž x 1 , x 2 , . . . , x n . s Ý m Ž . is1 x i , x 1 , . . . , x n y Ý is1 x 1 , x i , x 2 , . . . , Ž . x n . q ??? qŽy1. n Ý m is1 x 1 , . . . , x n , x i , for n G 1, Ž6. « Ž x . s 1 for all x g X, and Ž7. m Ž1. s x 1 q x 2 q ??? qx m .
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We refer the reader to Section 1 of wS1x for the proof that Ž CnŽ X; G ., n ; n g Z. is indeed an augmented, acyclic G-complex. We denote this complex by C?Ž X; G ., i.e., C?Ž X ; G . s Ž Cn Ž X ; G . , n ; n g Z . , where the lower dot reminds us that we are dealing with a chain complex. If A is a G-module, we denote C?Ž X ; G, A . s Ž C n Ž X ; G, A . , d n ; n g Z. , where the upper dot reminds us that we are dealing with a cochain complex, and C n Ž X; G, A. s Hom G Ž CnŽ X; G ., A. is the Z-complex, and dn s Hom G Ž nq1 , 1 A ., where 1 A indicates the identity homomorphism of the G-module A. C n Ž X; G, A. is called the nth cochain group. For all n g Z, the nth cohomology group of the Z-complex C?Ž X; G, A. is H n Ž X ; G, A . s
Ker Ž dn . Im Ž dny 1 .
s
Z n Ž X ; G, A . B n Ž X ; G, A .
.
Z n Ž X; G, A. is the nth cocycle group, and B n Ž X; G, A. is the nth coboundary group. 2. COHOMOLOGY GROUP IN DIMENSION y1 In order to study dy1 : Cy1 Ž X; G, A. ª C 0 Ž X; G, A., we must first obtain the expression for Cy1 Ž X ; G, A . s C 0 Ž X ; G, A . s Hom G Ž Z w X x , A . . Let T1 , T2 , . . . , Tu be the orbits of the permutation representation Ž G, X . Žsee wWix.. If x i s Ti for i s 1, 2, . . . , u, we refer to the set x 1 , x 2 , . . . , x u4 u as a set of representatives of the orbits Ti 4is1 . For each x g X, the subgroup of G whose elements leave x fixed is denoted by H Ž x .; i.e., H Ž x . s s g G N s x s x4 . The greatest common divisor of the indices w G: H Ž x .x is called the index of the permutation representation Ž G, X .. If x 1 , x 2 , . . . , x u4 is a set of
PERMUTATION REPRESENTATION COHOMOLOGY
159
u . representatives of Ž G, X . Žor representatives of the orbits Ti 4is1 , the study of Ž G, X . is equivalent to the simultaneous study of the classes of conjugate subgroups of G to which H Ž x 1 ., H Ž x 2 ., . . . , H Ž x u . belong. A cochain c g Hom G ŽZw X x, A. is completely determined by its values at x 1 , x 2 , . . . , x u , because, for any x g Ti , there exists s g G such that if s x i s x, then cŽ x . s cŽ s x i . s s cŽ x i .. Furthermore, for any s g H Ž x i ., if s x i s x i , then cŽ x i . s cŽ s x i . s s cŽ x i ., which implies that these values cŽ x i . cannot be chosen arbitrarily, but cŽ x i . g A H Ž x i . for i s 1, 2, . . . , u. We conclude:
PROPOSITION 2.1 ŽSee wS1x.. Cy1 Ž X ; G, A . s C 0 Ž X ; G, A . ( A H Ž x i . [ ??? [ A H Ž x u . , where [ designates the direct sum of Z-modules. If the action of G on A is tri¨ ial, then Cy1 Ž X; G, A. s C 0 Ž X; G, A. ( A [ ??? [ A, u times. We now ask: When is a y1-cochain c s Ž a1 , a2 , . . . , a u ., where a i s cŽ x i . g A H Ž x i . for i s 1, 2, . . . , u, a y1-cocycle? Knowing that for any x0 g X u
Ž dy1 c . Ž x 0 . s
Ý
u
SG N H Ž x i . c Ž x i . s
is1
Ý
SG N H Ž x i . Ž a i . ,
is1
where we use the customary symbol for the trace mapping SG N H Ž x i . : A H Ž x i . ª AG , we can easily derive that Z 0 Ž X; G, A. ( AG and B 0 Ž X; G, A. ( Ýuis1 SG N H Ž x i .Ž A H Ž x i . . Žsee wS1x.. We conclude: PROPOSITION 2.2. Zy1 Ž X; G, A. is isomorphic to the set of the u-tuples Ž a1 , a2 , . . . , a u ., where a i g A H Ž x i . for i s 1, 2, . . . , u and Ýuis1 SG N H Ž x i .Ž a i . s 0. If G acts tri¨ ially on A, then Zy1 Ž X; G, A. is isomorphic to the set of u-tuples Ž a1 , a2 , . . . , a u ., where a i g A for i s 1, 2, . . . , u, and Ýuis1w G: H Ž x i .x a i s 0. Now, we study By1 Ž X; G, A. when the permutation representation Ž G, X . is intransitive. As above, let the orbits of Ž G, X . be T1 , T2 , . . . , Tu , and T s u x 1 , x 2 , . . . , x u4 be a set of representatives of the orbits Ti 4is1 of Ž G, X .. We set H Ž xi . s s g G N s Ž xi . s xi4 ,
for x i g Ti , i s 1, 2, . . . , u,
and we let si j1 x j , si j2 x j , . . . , si jt i j x j be a set of representatives for the orbits of the permutation representation Ž H Ž x i ., Tj ., where Tj is one of the
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orbits of Ž G, X ., and x j Žin Tj . is the representation element of Tj , t i j is the number of double cosets H Ž x i .t H Ž x j . in G, where t g G, i, j s 1, 2, . . . , u, and si jk g G for k s 1, 2, . . . , t i j . PROPOSITION 2.3. The permutation representation Ž G, X 2 . has the pairs Ž x i , si jk ., for k s 1, 2, . . . , t i j , i, j s 1, 2, . . . , u, for a set of representati¨ es of its orbits, where X 2 is the Cartesian product of X with itself. Proof. We need to prove that:
s Ž x i , si jk x j . / Ž x i9 , si9 j9k 9 x j9 . for all s g G. We have three cases: Case 1. If i / i9, clearly s Ž x i . / x i9 for all s g G because x i and x i9 are not in the same orbit of Ž G, X .. Case 2. If i s i9 and j / j9, then we have
ssi jk x j / si j9k 9 x j9 , for all s g G, since x j and x j9 , j / j9, are in different orbits of Ž G, X .. Case 3. If i s i9, j s j9, but k / k9, we assume
s Ž x i , si jk x j . s Ž x i , si jk 9 x j . for some s g G. Then we know that s Ž x i . s x i , which means s must be in H Ž x i . and ssi jk x j s si jk 9 x j , which is impossible because si jk x j and si jk 9 x j , k / k9, are in different orbits of Ž H Ž x i ., Tj .. Now, consider an arbitrary pair Ž s x i , t x j . of X 2 , where s , t g G; then we have Ž x i , sy1t x j . s sy1 Ž s x i , t x j .. Since sy1t x j g Tj , there exists a r g H Ž x i . such that rsy1t x j s si jk x j for some k Ž1 F k F t i j ., then Ž x i , si jk x j . s rsy1 Ž s x i , t x j ., and the proposition is proved. According to Propositions 2.1 and 2.2, a cochain, c g Cy2 Ž X; G, A. s Hom G ŽZw X 2 x, A., is completely determined by its value at the pairs Ž x i , si jk x j .. These values cŽ x i , si jk x j . cannot be chosen arbitrarily, but cŽ x i , si jk x j . g A H i jk , where Hi jk s H Ž x i . l si jk H Ž x j . sy1 i jk . Since the cochain c is a G-homomorphism and if s Ž x i , si jk x j . s Ž x i , si jk x j . for some s g G, we must have c Ž s Ž x i , si jk x j . . s s c Ž x i , si jk x j . s c Ž x i , si jk x j . ,
PERMUTATION REPRESENTATION COHOMOLOGY
161
it implies that s Ž x i . s x i , ssi jk x j s si jk x j ; in other words, s g H Ž x i . and Ž . Ž . y1 s g si jk H Ž x j . sy1 i jk , i.e., s g H x i l si jk H x j si jk . We conclude ti j PROPOSITION 2.4. Cy2 Ž X; G, A. ( [i,u js1 [ks1 A H i jk , where A H i jk s a g A N s a s a for all s g Hi jk 4 .
From Proposition 2.4 we know that a cochain c g Cy2 Ž X; G, A. is completely determined by its values at the pairs Ž x i , si jk x j .; in particular, its values at the pairs Ž si jk x j , x i .. We like to know the relation between these pairs. Let us begin with the pair Ž si jk x j , x i ., we have
Ž si jk x j , x i . s si jk Ž x j , sy1 i jk x i . , there exists r g H Ž x j . such that
ry1sy1 i jk x i s sji k 9 x i
Ž 1.1.
for some k9 belonging to 1, 2, . . . , t ji 4 , and, obviously, k9 depends on ry1sy1 i jk , where si jk x i belongs to the set of representatives for the orbits of Ž H Ž x i ., Tj ., and sji k 9 x i belongs to the set of representatives for the orbits of Ž H Ž x j ., Ti .; therefore, we conclude, y1 y1 Ž si jk x j , x i . s si jk Ž x j , sy1 i jk x i . s si jk Ž r x j , rr si jk x i . y1 s si jk r Ž x j , ry1sy1 i jk x i . s sji k 9 Ž x j , sji k 9 x i . .
For convenience, we define the following mapping:
l : Ž x i , si jk x j . 4 ª Ž si jk x j , x i . 4 Ž . such that lŽ x i , si jk x j . s Ž si jk x j , x i . s sy1 ji k 9 x j , sji k 9 x i . If we set c Ž x i , si jk x j . s a i jk g A H i jk , c Ž x j , sji k 9 x i . s a ji k 9 g A H ji k 9 ; then
l a i jk s l c Ž x i , si jk x j . s c l Ž x i , si jk x j . s c Ž si jk x j , x j . s c sy1 ji k 9 Ž x j , sji k 9 x i . y1 s sy1 ji k 9 c Ž x j , sji k 9 x i . s sji k 9 a ji k 9 .
Ž 1.2.
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LEMMA 2.1.
l a i jk g A H i jk if and only if a ji k 9 g A H ji k 9.
Proof. a ji k 9 g A H ji k 9 m Ž H Ž x j . l sji k 9 H Ž x i . sy1 ji k 9 . a ji k 9 s a ji k 9 m Ž H Ž x j . sji k 9 l sji k 9 H Ž x i . . sy1 ji k 9 a ji k 9 s a ji k 9 y1 y1 m sy1 ji k 9 Ž H Ž x j . sji k 9 l sji k 9 H Ž x i . . sji k 9 a ji k 9 s sji k 9 a ji k 9 y1 y1 m Ž sy1 ji k 9 H Ž x j . sji k 9 l H Ž x i . .Ž sji k 9 a ji k 9 . s sji k 9 a ji k 9 ,
from Ž1.1. we have
ry1sy1 i jk x i s sji k 9 x i ,
r g HŽ xj.
i.e.,
ry1sy1 i jk s sji k 9 . y1 Ž ..Ž sy1 . Therefore, Ž si jk r H Ž x j . ry1sy1 i jk l H x i i jk 9 a ji k 9 s sji k 9 a ji k 9 , and from Ž1.2. we have
sy1 ji k 9 a ji k 9 s l a i jk , hence
Ž si jk H Ž x j . sy1 i jk l H Ž x i . . Ž l a i jk . s l a i jk , since r g H Ž x j ., r H Ž x j . ry1 s H Ž x j ., and Hi jk s H Ž x i . l si jk H Ž x j . sy1 i jk ; therefore, Hi jk Ž l a i jk . s l a i jk m l a i jk g A H i jk . Now we define the mapping ti j
u
Li:
[ [A js1 ks1
H i jk
ª AH Ž x i. ,
for i s 1, 2, . . . , u, by L i Ž a i jk N k s 1, 2, . . . , t i j ; j s 1, 2, . . . , u . u
s
ti j
Ý Ý
S H Ž x i .N H i jk Ž l a i jk y a i jk .
js1 ks1 u
s
ti j
Ý Ý js1 ks1
SH Ž x i .N H i jk w l y 1 x Ž a i jk . ,
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PERMUTATION REPRESENTATION COHOMOLOGY
where the mapping l is given in Ž1.2., 1 is the identity mapping, and SH Ž x i .N H i jk is the trace mapping from A H i jk to A H Ž x i ., and Ž a i jk N k s u ti j 1, 2, . . . , t i j ; j s 1, 2, . . . , u. belongs to [js1 [ks1 A H i jk . In short, we write ti j
u
Li s
SH Ž x i .N H i jk w l y 1 x
Ý Ý js1 ks1
for i s 1, 2, . . . , u. According to Lemma 2.1, we know that l a i jk g A H i jk , therefore L i is well-defined for i s 1, 2, . . . , u. Consider the following diagrams S 1 and S 2 : 6
Cy1 Ž X ; G, A . 6
S2
py2
py1
ti j
u
C 0 Ž X ; G, A . 6
S1
p0
u
dy2
A [ is1
u
H Ž xi.
6
H i jk
6
[ [A i , js1 ks1
d y1
6
dy2
Cy2 Ž X ; G, A . 6
dy1
A [ is1
H Ž xi.
where the mapping dy2 is defined by u
dy2 s
L; [ is1 i
this means that for any ti j
u
Ž ai jk N k s 1, 2, . . . , t i j ; i , j s 1, 2, . . . , u . g
[ [A
H i jk
,
i , js1 ks1
dy2 Ž a i jk N k s 1, 2, . . . , t i j ; i , j s 1, 2, . . . , u . u
s s
L Ža [ is1 i
ž
t1 j
u
Ý Ý
i jk
N k s 1, 2, . . . , t i j ; j s 1, 2, . . . , u .
S H Ž x 1 .N H 1 jk w l y 1 x Ž a1 jk . ,
js1 ks1
t2 j
u
Ý Ý
= w l y 1 x Ž a2 jk . , . . . ,
u
tu j
Ý Ý
SH Ž x u .N H u jk w l y 1 x Ž a u jk .
js1 ks1 u
s
u
ti j
[ Ý Ý is1
ž
S H Ž x 2 .N H 2 jk
js1 ks1
js1 ks1
S H Ž x i .N H i jk w l y 1 x Ž a i jk . .
/
/ Ž 1.3.
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The mapping py2 is defined by Py2 Ž a i jk N k s 1, 2, . . . , t i j ; i , j s 1, 2, . . . , u . Ž x i , si jk x j . s a i jk . The mappings dy2 and dy1 are defined as before. PROPOSITION 2.5. The diagram S 1 is commutati¨ e; in other words, dy1 ( py1 s po ( dy1 , where py1 , po are isomorphisms Ž see Proposition 2.1, actually p1 s p 2 . and defined by pj Ž Ž a1 , a2 , . . . , a u . . Ž x i . s a i ; j s y1, 0 and i s 1, 2, . . . , u. dy1 s D uŽÝuis1 SG N H Ž x i . ., where, D u is a diagonal mapping, D uŽ g . s Ž g, g, . . . , g ., u times. u
dy1 Ž Ž a1 , a2 , . . . , a u . . s D u
žÝ
SG N H Ž x i . Ž a i .
is1
u
s
žÝ
/ u
SG N H Ž x i . Ž a i . ,
is1
Ý
SG N H Ž x i . Ž a i . , . . . ,
is1 u
= Ý SG N H Ž x i . Ž a i . .
/
is1
Proof. For any Ž a1 , a2 , . . . , a u . g A H Ž x 1 . [ ??? [ A H Ž x u .
Ž dy1 ( py1 . Ž Ž a1 , a2 , . . . , au . . Ž x i . s py1 Ž Ž a1 , a2 , . . . , a u . . ( 0 Ž x i . s py1 Ž Ž a1 , a2 , . . . , a u . . Ž x 1 q x 2 q ??? qx m . u
s py1 Ž Ž a1 , a2 , . . . , a u . .
ž
Ý is1
SG N H Ž x i . Ž x i .
/
u
s
Ý
SG N H Ž x i . Ž py1 Ž Ž a1 , a2 , . . . , a u . . Ž x i . .
is1 u
s
Ý is1
SG N H Ž x i . Ž a i . .
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PERMUTATION REPRESENTATION COHOMOLOGY
On the other hand,
Ž p 0 ( dy1 . Ž Ž a1 , a2 , . . . , au . . Ž x i . s p 0 Ž dy1 Ž Ž a1 , a2 , . . . , a u . . . Ž x i . u
s p0
ž
Ý
u
SG N H Ž x i . Ž a i . , . . . ,
is1
/
SG N H Ž x i . Ž a i . Ž x i .
Ý is1
u
s
Ý
SG N H Ž x i . Ž a i . .
is1
Therefore, dy1 ( py1 s po ( dy1. THEOREM 2.1. The diagram S 2 is commutati¨ e. Proof. For any u t [ks1 Ž ai jk N k s 1, 2, . . . , t i j ; i , j s 1, 2, . . . , u . g [js1 ij
A H i jk ,
we set f Ž x i . s Ž dy2 ( py2 . Ž a i jk N k s 1, 2, . . . , t i j ; i , j s 1, 2, . . . , u . Ž x i . s py2 Ž a i jk N k s 1, 2, . . . , t i j ; i , j s 1, 2, . . . , u . ( y1 Ž x i . s py2 Ž a i jk N k s 1, 2, . . . , t i j ; i , j s 1, 2, . . . , u . =
žÝŽ x lgX
xl , xi . y
Ý Ž xi , xl .
x lgX
/
s py2 Ž a i jk N k s 1, 2, . . . , t i j ; i , j s 1, 2, . . . , u . =
ž
u
ti j
Ý Ý
ti j
u
SH Ž x i .N H i jkŽ si jk x j , x i . y
js1 ks1
Ý Ý
SH Ž x i .N H i jkŽ x i , si jk x j .
js1 ks1
s py2 Ž a i jk N k s 1, 2, . . . , t i j ; i , j s 1, 2, . . . , u . = u
s
ž
u
ti j
Ý Ý
SH Ž x i .N H i jk Ž lŽ x i , si jk x j . y Ž x i , si jk x j . .
js1 ks1 ti j
Ý Ý js1 ks1
u
SH Ž x i .N H i jk Ž l a i jk y a i jk . s
/
ti j
Ý Ý
SH Ž x i .N H i jk
js1 ks1
= w l y 1 x Ž a i jk . ,
/
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for i s 1, 2, . . . , u; on the other hand, we put g Ž x i . s Ž py1 ( dy2 . Ž a i jk N k s 1, 2, . . . , t i j ; i , j s 1, 2, . . . , u . Ž x i . , from Ž1.3. and the definition of py1 , u
g Ž x i . s py1 u
s
ž[ž is1 ti j
ti j
u
Ý Ý
SH Ž x i .N H i jk w l y 1 x Ž a i jk .
js1 ks1
//
Ž xi .
SH Ž x i .N H i jk w l y 1 x Ž a i jk . .
Ý Ý js1 ks1
Hence, f Ž x i . s g Ž x i ., for all i s 1, 2, . . . , u, which implies that f s g; i.e., py1 ( dy2 s dy2 ( py2 . THEOREM 2.2. By1 Ž X ; G, A . ( Im dy2 s u
s
u
Im L [ is1
u
t it
[ Ý Ý is1
ž
i
SH Ž x i .N H i jk w l y 1 x Ž A H i jk . .
js1 ks1
/
Proof. Since S 2 is commutative, py2 and py1 are isomorphisms, and the theorem follows. COROLLARY 2.1.
dy1 ( dy2 s 0.
Proof. Since dy1 ( dy2 s 0, and S1 and S 2 are commutative,
dy1 ( Ž dy2 ( py2 . s dy1 ( Ž py1 ( dy2 . , thus
dy1 ( dy2 ( py2 s Ž p 0 ( dy1 . ( dy2 , and dy1 ( dy2 s 0 since py2 and p 0 are isomorphisms. Remark 2.1. The ‘‘trace mappings’’ dy1 and dy2 Žbecause dy1 and dy2 consist of trace mappings. depend on the choice of representatives
167
PERMUTATION REPRESENTATION COHOMOLOGY
x 1 , x 2 , . . . , x u of Ž G, X . and the pairs Ž x i , si jk x j . N k s 1, 2, . . . , t i j ; i, j s 1, 2, . . . , u4 . We may henceforth say that dy1 and dy2 are determined up to isomorphisms. The computation of By1 Ž X; G, A., in the case that Ž G, X . is transitive, is given on page 69 of wAdx and in Section 11 of wS3x. The following theorem is an immediate consequence of Proposition 2.2, Theorem 2.2, and the definition of dy1 Žsee Proposition 2.5.. THEOREM 2.3. Hy1 Ž X ; G, A . (
Ž a1 , a2 , . . . , au . N Ýujs1 SG N H Ž x . Ž ai . s 0 4 i
[
u is1
ž
Ýujs1
ij Ýtks1
SH Ž x i .N H i jk w l y 1 x Ž A H i jk .
.
/
Ž For Ž G, X . transiti¨ e, see Theorem 4.3 in w Ad x..
3. COHOMOLOGY OF PERMUTATION REPRESENTATION IN NEGATIVE DIMENSIONS In the previous section, we discussed cohomology in dimension y1; now we use the induction method to study more general situations. For convenience, we adopt the following notations: I1 is a set of representatives of orbits for permutation representation Ž G, X ., I1 : X. Ž x i1 , x i2 . N x i1 g I1 , x i2 g I2 4 is a set of representatives of orbits for Ž G, X 2 ., where I1 : I2 : X; from Proposition 2.3, the pairs Ž x i , si jk x j ., for si jk g G, k s 1, 2, . . . , t i j , i, j s 1, 2, . . . , u, form a representative set for the orbits of Ž G, X 2 .; x i , x j g I1. We put I2 s si jk x j N k s 1, 2, . . . , t i j ; i , j s 1, 2, . . . , u4 and, without loss of generality, we can always choose si j1 x j s x j , i.e., si j1 s 1 g G. We assume that if Ž x i1 , x i2 , . . . , x i q . N x i1 g I1 , x i2 g I2 , . . . , x i q g Iq , I1 : I2 : ??? : Iq : X 4 is a representative set of orbits for Ž G, X q ., then there exists Iqq 1 such that Iq : Iqq1 : X, and Ž x i1 , x i2 , . . . , x i q , x i qq1 . N x i j g I j , I j : I jq1 , j s 1, 2, . . . , q, and x i qq1 g Iqq1 4 is a representative set of orbits for Ž G, X qq 1 .. The construction of Iqq 1 is as follows. Let H Ž x i1 , x i2 , . . . , x i q . s s g G N s Ž x i1 , x i2 , . . . , x i q . s Ž x i1 , x i2 , . . . , x i q .4 . Clearly, H Ž x i1 , x i2 , . . . , x i q . s H Ž x i1 . l H Ž x i2 . l ??? l H Ž x i q .. Let the set x i1, i2, . . . , i q, k N k s 1, 2, . . . , t i1, i2, . . . , i q,4 : X be a repre-
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sentative set of orbits for Ž H Ž x i1 , x i2 , . . . , x i q ., X . where t i1, i2, . . . , i q is the number of orbits of Ž H Ž x i1 , x i2 , . . . , x i q ., X .. Since k G 1, we can choose x i1, i2, . . . , i q, 1 s x i q for some x i1 , x i2 , . . . , and x i q . Hence, Iqq1 s Dqjs1 Dx i j g I j x i1, i2, . . . , i q, k N k s 1, 2, . . . , t i1, i2, . . . , i q,4 , and by the construction, Iq : Iqq1 : X. LEMMA 3.1. Ž x i1 , x i2 , . . . , x i q , x i qq1 . N x i j g I j , j s 1, 2, . . . , q q 14 is a representati¨ e set of orbits for Ž G, X qq 1 .. Proof. We need to prove that for any s g G,
s Ž x i1 , x i2 , . . . , x i q , x i qq1 . / Ž x i19 , x i2 9 , . . . , x i q9 , x i qq19 . , where x i j , x i j9 g I j , j s 1, 2, . . . , q q 1. If Ž x i1 , x i2 , . . . , x i q . / Ž x i19 , x i2 9 , . . . , x i q9 ., then for all s g G, s Ž x i1 , x i2 , . . . , x i q , x i qq1 . / Ž x i19 , x i2 9 , . . . , x i q9 , x i qq19 . because Ž x i1 , x i2 , . . . , x i q . and Ž x i19 , x i2 9 , . . . , x i q9 . are different orbits of Ž G, X q .. If Ž x i1 , x i2 , . . . , x i q . s Ž x i19 , x i2 9 , . . . , x i q9 ., and x i qq1 / x i qq19 , suppose that there exists a s g G such that
s Ž x i1 , x i2 , . . . , x i q , x i qq1 . s Ž x i1 , x i2 , . . . , x i q , x i qq19 . , which implies that s g H Ž x i1 , x i2 , . . . , x i q . and s x i qq1 s x i qq19 , x i qq1 , x i qq19 g Iqq1 , which is impossible. For any Ž x 1 , x 2 , . . . , x q , x qq1 ., x j g X, j s 1, 2, . . . , q q 1, because Ž x 1 , x 2 , . . . , x q . g X q there exists s g G such that
s Ž x 1 , x 2 , . . . , x q , x qq1 . s Ž x i1 , x i2 , . . . , x i q , s x qq1 . , where x i j g I j , j s 1, 2, . . . , q. Furthermore, there exists H Ž x i1 , x i2 , . . . , x i q . such that rs x qq1 s x i qq1 g Iqq1. Putting the above together, we have
rs Ž x 1 , x 2 , . . . , x q , x qq1 . s r Ž x i1 , x i2 , . . . , x i q , s x qq1 . s Ž x i1 , x i2 , . . . , x i q , rs x qq1 . s Ž x i1 , x i2 , . . . , x i q , x i qq1 . . Therefore, the lemma is proved.
rg
PERMUTATION REPRESENTATION COHOMOLOGY
169
It follows from the above lemma that we have PROPOSITION 3.1. q
Cyq Ž X ; G, A . (
[ [A js1 x gI ij
H Ž x i1 , x i2 , . . . , x i q .
,
j
where A H Ž x i1 , x i2 , . . . , x i q . s a g A N s a s a for all s g H Ž x i1 , x i2 , . . . , x i q .4 . Next we define a family of mappings qq 1 l k N k s 0, 1, . . . , q4 on X qq 1. qq 1 l k
Ž x i1 , x i2 , . . . , x i q , x i qq1 . s Ž x i1 , x i2 , . . . , x i k , x i qq1 , x i kq1 , . . . , x i q . ,
then there exists s g H Ž x i1 , x i2 , . . . , x i k . such that
Ž x i1 , x i2 , . . . , x i k , x i qq1 , x i kq1 , . . . , x i q . s s Ž x i1 , x i2 , . . . , x i k , x i kq10 , . . . , x i q0 , x i qq10 . , where x i j g I j , j s 1, 2, . . . , q, q q 1, and x i j0 g I j , j s k q 1, . . . , q q 1; i.e., qq 1 l k
Ž x i1 , x i2 , . . . , x i q , x i qq1 .
s s Ž x i1 , x i2 , . . . , x i k , x i kq10 , . . . , x i q0 , x i qq10 . . For any c s CyŽ qq1. Ž X; G, A., obviously, qq 1 l k
Ž c Ž x i1 , x i2 , . . . , x i q , x i qq1 . .
s s c Ž x i1 , x i2 , . . . , x i k , x i kq10 , . . . , x i q0 , x i qq10 . . Therefore,
qq 1 l k
is uniquely determined by the value of c on the set
Ž x i1 , x i2 , . . . , x i q , x i qq1 . N x i j g Ij , j s 1, 2, . . . , q q 1 4 . For all x i qq1 g Iqq1 , we assume that c Ž x i1 , x i2 , . . . , x i q , x i qq1 . s a i1, i2, . . . , i qq1 and c Ž x i1 , x i2 , . . . , x i k , x i kq10 , . . . , x i q0 , x i qq10 . s a i1, i2, . . . , i k , i kq10 , . . . , i qq10 , then qq 1 l k
Ž ai1, i2, . . . , i qq1 . s s ai1, i2, . . . , i k , i kq10 , . . . , i qq10 ,
for some s g H Ž x i1 , x i2 , . . . , x i k ..
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LEMMA 3.2.
qq 1 l k
Ž a i1, i2, . . . , i qq1 . g A H Ž x i1 , x i2 , . . . , x i qq 1 . if and only if
a i1, i2, . . . , i k , i kq10 , . . . , i qq10 g A H Ž x i1 , x i2 , . . . , x i k , x i kq 10 , . . . , x i q 0 , x i qq 10 . , k s 0, 1, . . . , q. Proof. Because a i1, i2, . . . , i k , i kq10 , . . . , i qq10 g A H Ž x i1 , x i2 , . . . , x i k , x i kq 10 , . . . , x i q 0 , x i qq 10 . m H Ž x i1 , x i2 , . . . , x i k , x i kq10 , . . . , x i q0 , x i qq10 . = a i1, i2, . . . , i k , i kq10 , . . . , i qq10 s a i1, i2, . . . , i k , i kq10 , . . . , i qq10 m s H Ž x i1 , x i2 , . . . , x i k , x i kq10 , . . . , x i q0 , x i qq10 . = sy1 Ž s a i1, i2, . . . , i k , i kq10 , . . . , i qq10 . s s a i1, i2, . . . , i k , i kq10 , . . . , i qq10 m s H Ž x i1 , x i2 , . . . , x i k , x i kq10 , . . . , x i q0 , x i qq10 . = sy1 Ž qq 1 l k Ž a i1, i2, . . . , i qq1 . . sqq 1 l k Ž a i1, i2, . . . , i qq1 . . Now we only need to prove that
s H Ž x i1 , x i2 , . . . , x i k , x i kq10 , . . . , x i q0 , x i qq10 . sy1 s H Ž x i1 , x i2 , . . . , x i q , x i qq1 . . Indeed,
s H Ž x i1 , x i2 , . . . , x i k , x i kq10 , . . . , x i q0 , x i qq10 . sy1 s s H Ž x i1 . sy1 l s H Ž x i2 . sy1 l ??? l s H Ž x i k . sy1 l s H Ž x i kq10 . sy1 l ??? l s H Ž x i qq10 . sy1 s H Ž x i1 . l H Ž x i2 . l ??? l H Ž x i k . l H Ž x i qq1 . l H Ž x i kq1 . l ??? l H Ž x i q . s H Ž x i1 , x i2 , . . . , x i k , x i kq1 , . . . , x i q , x i qq1 . because s x i kq10 s x i qq1 , s x i kq2 0 s x i kq1 , . . . , s x i qq10 s x i q . It implies that qq 1 l k Ž a i1, i2, . . . , i qq1 . g A H Ž x i1 , x i2 , . . . , x i qq 1 ..
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PERMUTATION REPRESENTATION COHOMOLOGY
We now define the following mapping: L i1, i2, . . . , i q :
[
x iqq1 gI qq1
A H Ž x i1 , x i2 , . . . , x i q , x i qq 1 . ª A H Ž x i1 , . . . , x i q .
by L i1, i2, . . . , i q Ž a i1 , i2, . . . , i qq1 N x i qq1 g Iqq1 . s
Ý
SH Ž x i1 , x i2 , . . . , x i q .N H Ž x i1 , x i2 , . . . , x i q , x i qq 1 .
qq 1 l 0
yqq1 l1 q ??? q Ž y1 .
x i qq1 gI qq1
=
q
qq1 l q
Ž ai1, i2, . . . , i qq1 . .
In short, we write L i1 , i2, . . . , i q s
Ý
SH Ž x i1 , x i2 , . . . , x i q .N H Ž x i1 , x i2 , . . . , x i q , x i qq 1 .
qq 1 l 0
yqq1 l1 q ??? q Ž y1 .
x i qq1 gI qq1
=
q
qq1 l q
,
for x i j g I j , j s 1, 2, . . . , q. According to Lemma 3.2, we know that L i1, i2, . . . , i q is well defined. THEOREM 3.1. The following diagram is commutati¨ e: dyŽ qq1.
Cyq Ž X ; G, A . 6
6
CyŽ qq1. Ž X ; G, A . 6 pyŽ qq1.
pyq
qq1
ij
H Ž x 1 , x 2 , . . . , x i q , x i qq 1 .
dyŽ qq1.
j
q where dyŽ qq1. s [js1
[x
i j g Ij
[ [A js1 x gI
6
[ [A js1 x gI
q
ij
H Ž x1 , x 2 , . . . , x i q .
j
L i1, i2, . . . , i q .
Proof. For any a i1, i2, . . . , i qq1 g A H Ž x i1 , x i2 , . . . , x i qq 1 ., we compute
Ž Ž dyŽ qq1. ( pyŽ qq1. .Ž ai1, i2, . . . , i qq1 N x i j g Ij , j s 1, 2, . . . , q q 1. . = Ž x i1 , x i2 , . . . , x i q . s Ž pyŽ qq1. Ž a i1, i2, . . . , i qq1 N x i j g I j , j s 1, 2, . . . , q q 1 . . =( yq Ž x i1 , x i2 , . . . , x i q . s Ž pyŽ qq1. Ž a i1, i2, . . . , i qq1 N x i j g I j , j s 1, 2, . . . , q q 1 . .
,
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=
ž Ý Ž x, x , x i1
i2 , . . . ,
xiq . y
Ý Ž x i1 , x, x i2 , . . . , x i q .
xgX
xgX q
q ??? q Ž y1 . Ž x i1 , x i2 , . . . , x i q , x .
/
s Ž pyŽ qq1. Ž a i1, i2, . . . , i qq1 N x i j g I j , j s 1, 2, . . . , q, q q 1 . .
ž
Ý
x i qq1 gI qq1
y
Ž SH Ž x
i1 ,
x i2 , . . . , x i q .N H Ž x i1 , x i2 , . . . , x i qq 1 .
Ž SH Ž x
Ý
x iqq1 gI qq1
q ??? q Ž y1 .
x i2 , . . . , x i q .N H Ž x i1 , x i2 , . . . , x i qq 1 .
i1 ,
q
Ž x i qq1 , x i1 , x i2 , . . . , x i q . .
Ý
x iqq1 gI qq1
Ž SH Ž x
i1 ,
Ž x i1 , x i qq1 x i2 , . . . , x i q . .
x i2 , . . . , x i q .N H Ž x i1 , x i2 , . . . , x i qq 1 .
= Ž x i1 , x i2 , . . . , x i q , x i qq1 . .
/
s Ž pyŽ qq1. Ž a i1 , i2, . . . , i qq1 N x i j g I j , j s 1, 2, . . . , q, q q 1 . .
ž
Ý
x iqq1 gI qq1
žS
H Ž x i1 , x i2 , . . . , x i q .N H Ž x i1 , x i2 , . . . , x i qq 1 .
=
qq 1 l 0
yqq1 l1 q ??? q Ž y1 .
= Ž x i1 , x i2 , . . . , x i q , x i qq1 . s
Ý
x iqq1 gI qq1
žS
q
qq1 l q
//
H Ž x i1 , x i2 , . . . , x i q .N H Ž x i1 , x i2 , . . . , x i qq 1 .
=
qq 1 l 0
yqq1 l1 q ??? q Ž y1 .
= Ž a i1, i2, . . . , i qq1 .
q
qq1 l q
/
On the other hand, we have
Ž Ž pyq ( dyŽ qq1. .Ž ai1, i2, . . . , i qq1 N x i j g Ij , j s 1, 2, . . . , q q 1. . = Ž x i1 , x i2 , . . . , x i q . q
s pyq
ž[ Ý js1
x i jgI j
L i1, i2, . . . , i q Ž a i1, i2, . . . , i qq1 N x i qq1 g Iqq1 .
= Ž x i1 , x i2 , . . . , x i q .
/
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PERMUTATION REPRESENTATION COHOMOLOGY
s L i1, i2, . . . , i q Ž a i1, i2, . . . , i qq1 N x i qq1 g Iqq1 . s
Ý
x iqq1 gI qq1
žS =
H Ž x i1 , x i2 , . . . , x i q .N H Ž x i1 , x i2 , . . . , x i qq 1 .
qq 1 l 0
yqq1 l1 q ??? q Ž y1 .
q
qq1 l q
Ž ai1, i2, . . . , i qq1 . .
/
Therefore, we have proved the theorem. COROLLARY.
dyq ( dyŽ qq1. s 0.
Proof. It follows from dyq ( dyŽ qq1. s 0. THEOREM 3.2. Zyq Ž X ; G, A . (
½Ž
a i1 , i2, . . . , i q N x i j g I j , j s 1, 2, . . . , q . N a i1, i2, . . . , i q
g A H Ž x i1 , x i2 , . . . , x i q . ,
Ý
x iqgI q
S H Ž x i1 , x i2 , . . . , x i qy 1 .N H Ž x i1 , x i2 , . . . , x i q .
= q l0 yq l1 q ??? q Ž y1 .
qy 1
q l qy1
Ž ai1, i2, . . . , i q . s 0
5
for x i j g I j , j s 1, 2, . . . , q y 1 . q
Byq Ž X ; G, A . (
[ [ js1 x gI ij
S H Ž x i1 , x i2 , . . . , x i q .N H Ž x i1 , x i2 , . . . , x i qq 1 .
Ý
j
x i qq1 gI qq1
=
qq 1 l 0
yqq1 l1 q ??? q Ž y1 .
q
qq1 l q
= Ž A H Ž x i1 , x i2 , . . . , x i1 , x i qq 1 . . . In short, Hyq Ž X ; G, A . ( Ker dyq rIm dyŽ qq1. qy1
s
[ [ Ker Ž L js1 x gI ij
q
i1, i2, . . . , i qy1
.
j
[ [ Im Ž L js1 x gI ij
i1, i2, . . . , i q
..
j
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