Conjugacy classes in parabolic subgroups of general linear groups

TL;DR

This paper establishes a formula linking unipotent conjugacy classes in parabolic subgroups of finite general linear groups with those in smaller dimensions, enabling calculations of conjugacy classes in specific matrix groups.

Contribution

It generalizes a formula connecting conjugacy classes across different subgroup dimensions and provides methods to compute these classes in small cases.

Findings

Derived a formula relating conjugacy classes in parabolic subgroups to smaller dimensions

Expressed the number of conjugacy classes of unitriangular matrices in terms of smaller parabolic subgroups

Provided methods for calculating unipotent conjugacy classes in small-dimensional parabolic subgroups

Abstract

We prove a formula connecting the number of unipotent conjugacy classes in a maximal parabolic subgroup of a finite general linear group with the numbers of unipotent conjugacy classes in various parabolic subgroups in smaller dimensions. We generalise this formula and deduce a number of corollaries; in particular, we express the number of conjugacy classes of unitriangular matrices over a finite field in terms of the numbers of unipotent conjugacy classes in maximal parabolic subgroups over the same field. We show how the numbers of unipotent conjugacy classes in parabolic subgroups of small dimensions may be calculated.