Preliminaries
CHAPTER 1. PRELIMINARIES 30
[d, g]u u[d, g]u-1
~gu(d-l)u(g-l)u
~gu(~)-l(gu)-l
= [~,gU] E Im(c/rg),
that is Im(<jJg) is Cy invariant. Now since Nand N/Ker(c/rg) ( C;:,! Im(c/rg)) are Cyinvariant, we conclude that Ker(c/rg) is Cy invariant. 0
We see that N acts on Ng by conjugation and K = Im(c/rg) acts on Ng by left multiplication such that the resulting orbits of the two actions are the same. Hence the action ofCgon the orbits ofN acting on Ng is the same as the action ofOgon the module N / K. Thus the orbits of the action of K on Ng can be identified with the elements ofN / K. Let (}i EIrr(N),'lfJi EIrr(Hd and 'lfJi be an extension of(}i to Hi. Then 'lfJi is constant on the orbits ofN acting on Ng. So we may define a class functionJ.Lon N / K by J.L(Kdjg) = 'lfJi(djg), wheredj EN, djgE Qj is a representative of the j-th orbit ofN acting on Ng and d1 = IN. Then J.L(Kg) = 'lfJi(g). Let
p,
be an extension ofJ.L to the inertia group ofJ.L in Cg. Then induction ofp,
to G evaluated on the elements ofNg is equivalent to the induction ofp,
to Cg/K evaluated on the elements of N / K. However for G a non-split extension, it may happen that J.L is not a character of N / K. In that case eJ.L will be a character of N / K, wheree
is anappropriate p-th root of unity.
Proposition 1.5.2 Let N be an elementary abelian p-group. Let9E G and dEN.
Define an endomorphism c/rg of N by </Jg(d) = d(d-1)g, with image K = Im(</Jg).
(i) IfGis a split extension, then the Fischer-ClifJord matrix at a non-identity coset of N in G is the matrix of orbit sums of Og acting on the rows of the character table of N / K with duplicate columns discarded.
(ii) IfG is a non-split extension, then the Fischer-ClifJord matrix is the matrix of orbit sums of Cg acting on the rows of the character table of N / K with duplicate columns discarded and with each row multiplied by an appropriate p-th root of unity.
Proof. See Lemma 5.3 in [18] or page 119 in [35]. 0
CHAPTER 1. PRELIMINARIES
1.5.2 Properties of Fischer-Clifford Matrices
31
The properties of Fischer-Clifford matrices have been discussed in [1], [35], [36], [47], [51], [65], we also discuss them in this subsection. Let K be a group and A :S Aut(K).
Then by Theorem 1.4.4 the group A acts on the conjugacy classes of elements of K and on the irreducible characters ofK resulting in the same number of orbits.
Lemma 1.5.3 Suppose we have the following matrix describing the above actions:
1
=
II l2 lj ItSI 1 1 1 1
S2 a21 a22 a2j a2t
Si ail ai2 aij ait
St atl at2 atj att
where alj = 1 for j E {I, 2, ... ,t}, lj's are lengths of orbits of A on the conjugacy classes of K, Si'S are lengths of orbits of A on Irr(K) and aij is the sum of Si ir- reducible characters of K on the element Xj, where Xj is an element of the orbit of length lj. Then the following relation holds for i,i' E {I, 2, ... ,t}:
L
t aijai1jlj = IKlsibiil • j=lProof. See Lemma 4.2.2 in [65]. 0
Let Xj E X(g) and define mj = [Og :0C(Xj)]. The Fischer-Clifford matrix M(g) is partitioned row-wise into blocks, where each block corresponds to an inertia group.
The columns of M(g) are indexed by X(g) and to each Xj E X(g), corresponds ICc(Xj)
I
of a conjugacy class of G. The rows of M(g) are indexed by R(g) and on the left of each row we write jCHi(Yk)l, where Yk fuses into [g] in G. The following result gives the orthogonality relation for M(g).Proposition 1.5.4 [65](Column orthogonality) Let G
=
N.G, then" IC ( )/
(i,ykl (i,Yk) - ~10 ( )1
LJ Hi Yk aj aj' - Ujj' C Xj (i,Yk)ER(g)
CHAPTER 1. PRELIMINARIES 32 Proof. See Proposition 4.2.3 in [65J. 0
Theorem 1.5.5 a)l,g) = 1for allj = {I, 2, ... ,c(g)}
Proposition 1.5.6 ([35), [65}, [47j) The matrixM(lc) is the matrix with rows equal to the orbit sums of the action ofG on Irr(N) with duplicate columns discarded. For this matrix we have a)i,lG) = [G :Hi], and an orthogonality relation for rows:
Proof. The (i,lc),jth entry ofM(lc) is given by
where we sum over representatives of conjugacy classes ofHi which fuse into [Xj] in G.
Therefore ay,lG) = 7J;r(Xj)' By Theorem 1.4.6 we have
7J;r
E Irr(G) and we obtain that ((7J;r)N,(}i)=
((7J;i)N,(}i)=
1. Therefore by Clifford's theorem (7J;r)N=
La(}a,where the summation is taken over all (}a E Irr(N) such that (}a is conjugate to (}i.
So for Xj EN we obtain that a)i,lG) =
La
(}a(Xj). The orthogonality relation follows by Lemma 1.5.3. 0Following from Lemma 1.5.3, Proposition 1.5.4 and the results proved by Fischer in [18], the Fischer-Clifford matrix M (g) satisfies the following properties:
(a) IX(g)1 = /R(g)f
( ) "C u(i,Yk)ER(g) aj(i,Yk) (i,Yk)ICaj' Hi Yk(
)1 -
- fJjj'10 (
G Xj)1
(d) M(g) is square and nonsingular.
CHAPTER 1. PRELIMINARiES
For N is elementary abelian, M(g) also satisfies the following (e) a(i,Yk) -1 - IGHi(Yk)!IGc(g)1
(f) la~i'Yk)1 ~ la)i'Yk)
I,
1<
j ~ c(g).33
Now let G = N:G be a split extension and N be an elementary abelian 2-group.
By Proposition 1.5.2 the Fischer-Clifford matrix M(g) is given by
1 1 1 1 1
d21 d 22 d 23 d 2j
M{g) =
dil d i2 d i3 d ij
dtl dt2 dt3 dtj
wheredij'S are the orbit sums of C9 acting on the rows of the character table ofNjK.
Proposition 1.5.7 ([47]) di1 E IN for alli E {2, 3, ... ,t}.
Proof. By Proposition 1.5.2, we obtain that
dil =
:L
X(lN/K)XE.6.i
where b.i's are the orbits of Cg acting on Irr(NjK). Since X(1N/K) = deg(x), we have d i1 E IN ViE {2, 3, ...,t}. 0
For j ~ 2, we obtain that
dij
= L
X(Xj)XE.6.i
where Xj E NjK is a representative of the j-th orbit under the action of Cg on the elements ofNjK. Since X(Xj) E{-I, I}, we have d ij E lL.