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

Marberg, Eric

Journal of Algebraic Combinatorics, Tome 35 (2012) no. 1, pp. 61-92.

Résumé

Summary: Let $U _{ n }$ denote the group of $n\times n$ unipotent upper-triangular matrices over a fixed finite field $\mathbb F _{ q}$ mathbbF_q, and let $U _{ P}$ U_mathcalP denote the pattern subgroup of $U _{ n }$ corresponding to the poset $P$ mathcalP. 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 $U _{ P}$ U_mathcalP and certain $\mathbb F _{ q} \mathbb $F_q-labeled subposets of $P$ mathcalP. This bijection generalizes the correspondence identified by André and Yan between the supercharacters of $U _{ n }$ and the $\mathbb F _{ q}$ mathbbF_q-labeled set partitions of ${1,2,\cdots , n}$. At present, few explicit descriptions appear in the literature of the superclasses and supercharacters of infinite families of algebra groups other than ${ U _{ n }: n\in \Bbb N}$. This work significantly expands the known set of examples in this regard.

Détail

Keywords: keywords unitriangular group, pattern groups, algebra groups, supercharacter theories, supercharacters, superclasses, labeled posets, labeled set partitions