Notes on p-blocks of characters of finite groups

Abstract

Let G be a finite group and let p be a fixed prime number. In this paper we shall study the numbers of irreducible characters in a p-block of G with an abelian defect group and the contributions of subsections to the inner product of irreducible characters of G. As is seen in [ 10, 111 Broue and Puig’s generalized characters, which are defined in [3], are relevant for the study of p-blocks with abelian defect groups. They will play an important role in this paper too. Let K be the algebraic number field containing the /Gi th roots of 1, let p be a prime ideal divisor of p in K, and let o be the ring of p-integers in K. Let B be a p-block and let (n, b) be a subsection associated with B. (n, b) is a pair such that ‘II is a p-element of G and b is a p-block of C(n) associated with B. In [3] (n, b) is called a (B, G)-Brauer element. Let x be an ordinary irreducible character in B. The function x”’ ‘) is the central function on G which vanishes outside the p-section of rc and which is such that x(X’ “(np) = x(npE,), where p is a p-regular element of C(n) and E, is the block idempotent of oC(n) corresponding to b. Following [I], for x, XI E B, we denote by m, xs (X3 ‘) the inner product of ~(“9 ‘) and x”“’ ‘) ~ rnpx?) is equal to the inner product of ~(~3 b, and 2’ by Brauer and &ma’s orthogonality relation. rnpxf) is called the contribution of (n, b) to the inner product (x, x’). By [l, (5B)],