Linear characters for blocks with cyclic defect

Abstract

We show that associated with each p-block B with non trivial cyclic defect group D of a finite group are two linear characters ${\alpha }_B$ and ${\mu }_B$ of a cyclic group E of order e. Here e is the number of non exceptional irreducible characters that B has. The first character ${\alpha }_B$ describes the action of E on D. The second character ${\mu }_B$ arises from certain field extensions. The order of ${\mu }_B$ is the p-local Schur index of any exceptional $\chi \in \mathrm{Irr}\left(B\right)$, giving an alternative description of this invariant to the one given by Benard [1]. From the relationship of ${\alpha }_B$ and ${\mu }_B$ we give an alternative description to the one given by Nebe [6] of the invariant associated with each exceptional $\chi \in \mathrm{Irr}\left(B\right)$.