Counting positive defect irreducible characters of a finite group

Abstract

Let z+ (G) be the number of ordinary irreducible characters of a finite group G which have positive defect with respect to a prime p. We express z+(G) as the p- adic limit of a sequence of enumerative parameters of G and p. When p = 2, and under a suitable hypothesis on the Sylow 2- subgroups of G, we give two local characterisations of the parity of z+(G), one of them compatible with Alperin’s Weight Conjecture, the other apparently independent.