Supercharacters and pattern subgroups in the upper triangular groups

TL;DR

This paper explores the relationship between supercharacters and pattern subgroups in the upper triangular groups over finite fields, providing a new correspondence that enhances understanding of their structure.

Contribution

It introduces a novel correspondence between supercharacters and specific pattern subgroups within upper triangular groups, expanding the structural analysis of these groups.

Findings

Established a correspondence between supercharacters and pattern subgroups.

Connected supercharacters with conjugation by monomial matrices.

Enhanced understanding of the subgroup structure in upper triangular groups.

Abstract

Let $U_n(q)$ denote the upper triangular group of degree $n$ over the finite field $\F_q$ with $q$ elements. It is known that irreducible constituents of supercharacters partition the set of all irreducible characters $\Irr(U_n(q)).$ In this paper we present a correspondence between supercharacters and pattern subgroups of the form $U_k(q)\cap {}^wU_k(q)$ where $w$ is a monomial matrix in $GL_k(q)$ for some $k<n.$