Calculating conjugacy classes in Sylow $p$-subgroups of finite Chevalley groups

TL;DR

This paper develops and implements an algorithm to compute conjugacy classes in Sylow p-subgroups of finite Chevalley groups, revealing polynomial patterns in the number of classes for groups of rank up to 6.

Contribution

It advances an existing algorithm, providing a practical implementation in GAP to compute conjugacy classes in these groups.

Findings

Number of conjugacy classes is polynomial in q for rank ≤ 6 groups

Algorithm successfully parametrizes conjugacy classes in Sylow p-subgroups

Implementation enables calculations for groups of rank up to 6

Abstract

In earlier work, the first author outlined an algorithm for calculating a parametrization of the conjugacy classes in a Sylow $p$-subgroup $U(q)$ of a finite Chevalley group $G(q)$, valid when $q$ is a power of a good prime for $G(q)$. In this paper we develop this algorithm and discuss an implementation in the computer algebra language {\sf GAP}. Using the resulting computer program we are able to calculate the parametrization of the conjugacy classes in $U(q)$, when $G(q)$ is of rank at most 6. In these cases, we observe that the number of conjugacy classes of $U(q)$ is given by a polynomial in $q$ with integer coefficients.