Conjugacy classes in maximal parabolic subgroups of general linear groups

TL;DR

This paper computes conjugacy classes in maximal parabolic subgroups of general linear groups by reducing the problem to matrix normal forms, providing explicit classifications for small dimensions and polynomial formulas over finite fields.

Contribution

It introduces a method to classify conjugacy classes in certain parabolic subgroups via matrix problems and provides explicit results for small cases and finite fields.

Findings

Conjugacy classes are classified for blocks with dimension less than 6.

Number of conjugacy classes over finite fields is polynomial in q.

Explicit normal forms are obtained for small-dimensional cases.

Abstract

We compute conjugacy classes in maximal parabolic subgroups of the general linear group. This computation proceeds by reducing to a ``matrix problem''. Such problems involve finding normal forms for matrices under a specified set of row and column operations. We solve the relevant matrix problem in small dimensional cases. This gives us all conjugacy classes in maximal parabolic subgroups over a perfect field when one of the two blocks has dimension less than 6. In particular, this includes every maximal parabolic subgroup of GL_n(k) for n < 12 and k a perfect field. If our field is finite of size q, we also show that the number of conjugacy classes, and so the number of characters, of these groups is a polynomial in $q$ with integral coefficients.