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

Abstract

Let U n denote the group of n×n unipotent upper-triangular matrices over a fixed finite field $\mathbb{F}_{q}$ , and let $U_{\mathcal{P}}$ denote the pattern subgroup of U n corresponding to the poset $\mathcal{P}$ . 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 $U_{\mathcal{P}}$ and certain $\mathbb {F}_{q}$ -labeled subposets of $\mathcal{P}$ . This bijection generalizes the correspondence identified by André and Yan between the supercharacters of U n and the $\mathbb{F}_{q}$ -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∈ℕ}. This work significantly expands the known set of examples in this regard.