Constructing characters of Sylow $p$-subgroups of finite Chevalley groups

TL;DR

This paper develops an algorithmic approach to classify and construct irreducible characters of Sylow p-subgroups in finite Chevalley groups, with a focus on type F4 and prime-specific variations.

Contribution

It introduces a reduction procedure and parametrization method for irreducible characters of Sylow p-subgroups, including explicit treatment for type F4 and prime-dependent differences.

Findings

Parametrization is uniform for good primes p > 3.

Parametrization differs for the bad prime p = 3.

Method has been applied to all groups of rank 4 or less.

Abstract

Let $q$ be a power of a prime $p$, let $G$ be a finite Chevalley group over $\mathbb{F}_q$ and let $U$ be a Sylow $p$-subgroup of $G$; we assume that $p$ is not a very bad prime for $G$. We explain a procedure of reduction of irreducible complex characters of $U$, which leads to an algorithm whose goal is to obtain a parametrization of the irreducible characters of $U$ along with a means to construct these characters as induced characters. A focus in this paper is determining the parametrization when $G$ is of type $\mathrm{F}_4$, where we observe that the parametrization is "uniform" over good primes $p > 3$, but differs for the bad prime $p = 3$. We also explain how it has been applied for all groups of rank $4$ or less.