A characterisation of nilpotent blocks

TL;DR

This paper characterizes nilpotent blocks of finite groups using character degrees and subgroup structures, extending Isaacs' results and involving hyperfocal subalgebras and fusion systems.

Contribution

It provides a new criterion for nilpotency of blocks based on the exact power of p dividing a sum of character degrees, linking it to subgroup and fusion system properties.

Findings

Characterization of nilpotent blocks via character degree sums

Connection between hyperfocal subalgebra characters and subgroup abelianness

A conjecture relating characters of hyperfocal subalgebras to subgroup structure

Abstract

Let $B$ be a $p$-block of a finite group, and set $m=$ $\sum \chi(1)^2$, the sum taken over all height zero characters of $B$. Motivated by a result of M. Isaacs characterising $p$-nilpotent finite groups in terms of character degrees, we show that $B$ is nilpotent if and only if the exact power of $p$ dividing $m$ is equal to the $p$-part of $|G:P|^2|P:R|$, where $P$ is a defect group of $B$ and where $R$ is the focal subgroup of $P$ with respect to a fusion system $\CF$ of $B$ on $P$. The proof involves the hyperfocal subalgebra $D$ of a source algebra of $B$. We conjecture that all ordinary irreducible characters of $D$ have degree prime to $p$ if and only if the $\CF$-hyperfocal subgroup of $P$ is abelian.