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

Let Un denote the group of n× n unipotent upper-triangular matrices over a fixed finite field Fq , and let UP denote the pattern subgroup of Un 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 Un. 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 UP and certain Fq -labeled subposets of P . This bijection generalizes the correspondence identified by André and Yan between the supercharacters of Un and the Fq -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 {Un : n ∈N}. This work significantly expands the known set of examples in this regard.