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

Let U _n denote the group of n×n unipotent upper-triangular matrices over a fixed finite field F _q , and let U _P denote the pattern subgroup of U _n corresponding to the poset 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 _P and certain F _q-labeled subposets of P. This bijection generalizes the correspondence identified by André and Yan between the supercharacters of U _n and the 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.