Unitriangular basic sets, Brauer characters and coprime actions

Feng, Zhicheng; Späth, Britta

doi:10.1090/ert/635

Unitriangular basic sets, Brauer characters and coprime actions
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by Zhicheng Feng and Britta Späth;

Represent. Theory 27 (2023), 115-148

DOI: https://doi.org/10.1090/ert/635

Published electronically: May 1, 2023

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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 Bonnafé–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 inductive Brauer–Glauberman condition that aims to establish a Glauberman correspondence for Brauer characters, given a coprime action.

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