The Brauer trees of non-crystallographic groups of Lie type

TL;DR

This paper determines the Brauer trees of unipotent blocks with cyclic defect groups in certain non-crystallographic groups of Lie type, confirming predictions from Broué's conjecture using geometric and algebraic methods.

Contribution

It explicitly constructs Brauer trees for specific non-crystallographic groups, extending the understanding of block theory in these cases.

Findings

Brauer trees for $I_2(n,q)$, $H_3(q)$, and $H_4(q)$ are determined.

Results are consistent with Broué's conjecture and Deligne--Lusztig theory.

The approach combines geometric and algebraic techniques for block analysis.

Abstract

In this article we determine the Brauer trees of the unipotent blocks with cyclic defect group in the `groups' $I_2(n,q)$, $H_3(q)$ and $H_4(q)$. The degrees of the unipotent characters of these objects were given by Lusztig, and using the general theory of perverse equivalences we can reconstruct the Brauer trees that would be consistent with Deligne--Lusztig theory and the geometric version of Brou\'e's conjecture. We construct the trees using standard arguments whenever possible, and check that the Brauer trees predicted by Brou\'e's conjecture are consistent with both the mathematics and philosophy of blocks with cyclic defect groups.