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

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.