Superclasses and supercharacters of normal pattern subgroups of the unipotent upper triangular matrix group

TL;DR

This paper classifies and describes superclasses and supercharacters of normal pattern subgroups of unipotent upper triangular matrices over finite fields, extending known correspondences and providing new explicit examples.

Contribution

It classifies all normal pattern subgroups of $U_n$, describes their superclasses and supercharacters, and establishes a bijection with labeled subposets, generalizing previous results.

Findings

Classified all normal pattern subgroups of $U_n$

Established a bijection between supercharacters and labeled subposets

Extended supercharacter theory to new classes of algebra groups

Abstract

Let $U_n$ denote the group of $n\times n$ unipotent upper-triangular matrices over a fixed finite field $\FF_q$, and let $U_\cP$ denote the pattern subgroup of $U_n$ corresponding to the poset $\cP$. This work examines the superclasses and supercharacters, as defined by Diaconis and Isaacs, of the family of normal pattern subgroups of $U_n$. After classifying all such subgroups, we describe an indexing set for their superclasses and supercharacters given by set partitions with some auxiliary data. We go on to establish a canonical bijection between the supercharacters of $U_\cP$ and certain $\FF_q$-labeled subposets of $\cP$. This bijection generalizes the correspondence identified by Andr\'e and Yan between the supercharacters of $U_n$ and the $\FF_q$-labeled set partitions of $\{1,2,...,n\}$. At present, few explicit descriptions appear in the literature of the superclasses and supercharacters of infinite families of algebra groups other than $\{U_n : n \in \NN\}$. This work signficantly expands the known set of examples in this regard.