Bounding the number of characters in a block of a finite group

TL;DR

This paper establishes a new, strong upper bound on the number of irreducible characters in a p-block of a finite group, based on local invariants and fusion system data, unifying previous bounds.

Contribution

It introduces a novel upper bound on k(B) that depends on local invariants and fusion system data, strengthening and unifying earlier bounds.

Findings

Provides a new upper bound on the number of irreducible characters in a p-block.

Unifies previous bounds by Brauer, Wada, and others.

Enhances understanding of the relationship between local invariants and character counts.

Abstract

We present a strong upper bound on the number k(B) of irreducible characters of a p-block B of a finite group G in terms of local invariants. More precisely, the bound depends on a chosen major B-subsection (u,b), its normalizer N_G(\langle u\rangle,b) in the fusion system and a weighted sum of the Cartan invariants of b. In this way we strengthen and unify previous bounds given by Brauer, Wada, K\"ulshammer--Wada, H\'ethelyi--K\"ulshammer--Sambale and the present author.