Characters of positive height in blocks of finite quasi-simple groups

TL;DR

This paper proves Eaton and Moretó's height conjecture for various classes of finite groups, including quasi-simple, reductive, and symmetric groups, by analyzing character degrees of Sylow subgroups.

Contribution

It confirms the conjecture for principal blocks of quasi-simple groups, all blocks of finite reductive groups in their defining characteristic, and covering groups of symmetric and alternating groups.

Findings

Confirmed the conjecture for principal blocks of quasi-simple groups.

Determined minimal non-trivial character degrees of Sylow p-subgroups.

Provided evidence for blocks of Lie type groups in cross characteristic.

Abstract

Eaton and Moret\'o proposed an extension of Brauer's famous height zero conjecture on blocks of finite groups to the case of non-abelian defect groups, which predicts the smallest non-zero height in such blocks in terms of local data. We show that their conjecture holds for principal blocks of quasi-simple groups, for all blocks of finite reductive groups in their defining characteristic, as well as for all covering groups of symmetric and alternating groups. For the proof, we determine the minimal non-trivial character degrees of Sylow $p$-subgroups of finite reductive groups in characteristic~$p$. We provide some further evidence for blocks of groups of Lie type considered in cross characteristic.