Local Schur indices, blocks, and character values

Abstract

It is shown that ordinary irreducible characters of height zero in 2-blocks of finite groups have trivial 2-local Schur index. In view of Brauer's height zero conjecture, it is therefore extremely plausible that the irreducible characters in 2-blocks with abelian defect groups have trivial 2-local Schur index. If q is an odd prime, then there is no relationship expected between the height of an irreducible character in a q-block and the q-local Schur index, but a condition for a prime p≠q not to divide this index is given in terms of a rationality property of a defect class of the character's p-block.