Towards a supercharacter theory of the parabolic subgroups

TL;DR

This paper develops a supercharacter theory for certain parabolic subgroups of GL(n, q), focusing on those with block sizes up to two, and proposes hypotheses for extending this theory to all parabolic subgroups.

Contribution

It introduces a supercharacter theory for specific parabolic subgroups and hypothesizes a general framework for all such subgroups in GL(n, q).

Findings

Supercharacter theory constructed for parabolic subgroups with blocks ≤ 2.

Formulation of hypotheses for general parabolic subgroups.

Foundation for future extension of supercharacter theory.

Abstract

The supercharacter theory is constructed for the parabolic subgroups of $\mathrm{GL}(n,\Fq)$ with blocks of orders less or equal to two. The author formulated the hypotheses on construction of a supercharacter theory for an arbitrary parabolic subgroup in $\mathrm{GL}(n,\Fq)$.