Symmetry And Group

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By Makhnev A. A.

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I'm grateful to Burkhard Kiilshammer who pointed out to me that the answer is given in a paper of Brauer and Feit. 42 R. 1. Theorem (cf. [BFD. For X E Irr(G) we have ld(x) . e~ . X E R(G) ld(x)-I. e~ . e. ld(x) . x(l)[ = IGI[. In particular, for n E No we have In . 5). Proof This theorem is not stated explicitly in [BF]. There, the integers . = l d [' . (Xi,ee' Xj), 1::::: i,j::::: m, are studied, where XI, ... ,Xm are the irreducible characters in an l-block of G of defect d. It is easy to see that for X E Irr( G) the irreducible constituents of e~.

5 an explicit formula for a~(x), H ::: G, X E R(H), and it suffices to prove that I(N H (er)/Ho)1' I H INH (er) / Ho I mrpll (res HIl (X)) is an integer for any H ::: G, any chain a = ((Ho,

There is a one to one correspondence between all characters 1/1 E Irr( N E(C)[~J) lying over ep and all characters 1/1' E Irr( E[C, ep, ~]). Furthermore, the number of such 1/1 with a given defect can be computed from the defect of ep and the cohomology class a [C, ep, ~] E H 2 ( NE (C , ep), U (~) ) corresponding to the extension E[C, ep, ~]. In addition to the above d, B [~] and F, we now fix an element a in the cohomology group H 2 (F, u (~»). 9. For any p-chain C of G we denote by k(C, B[~], d, F, a) the number of irreducible projective ~-characters ep of NG(C)[~] such that deep) = d, B(cP)G[J] = B[~], N E(C, ep) = F and a[C, cP,~] = a.

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