Unitriangular basic sets, Brauer characters and coprime actions

TL;DR

This paper proves that certain blocks of finite groups of Lie type have unitriangular decomposition matrices, using properties of normal subgroups, character extensions, and recent results on unipotent blocks, aiding the study of automorphisms on Brauer characters.

Contribution

It establishes conditions under which the decomposition matrix of a group is unitriangular, extending recent results to blocks of quasi-simple groups of Lie type and applying this to automorphism actions.

Findings

Blocks of quasi-simple groups of Lie type have unitriangular decomposition matrices under certain conditions.

The inductive Brauer--Glaubermann condition is verified for coprime actions on finite groups.

Unitriangularity of decomposition matrices is linked to character extension properties and group structure.

Abstract

We show that the decomposition matrix of a given group $G$ is unitriangular, whenever $G$ has a normal subgroup $N$ such that the decomposition matrix of $N$ is unitriangular, $G/N$ is abelian and certain characters of $N$ extend to their stabilizer in $G$. Using the recent result by Brunat--Dudas--Taylor establishing that unipotent blocks have a unitriangular decomposition matrix, this allows us to prove that blocks of groups of quasi-simple groups of Lie type have a unitriangular decomposition matrix, whenever they are related via Bonnaf\'e--Dat--Rouquier's equivalence to a unipotent block. This is then applied to study the action of automorphisms on Brauer characters of finite quasi-simple groups. We use it to verify the so-called {\it inductive Brauer--Glaubermann condition}, that aims to establish a Glauberman correspondence for Brauer characters, given a coprime action.