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.
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Marberg, E. Superclasses and supercharacters of normal pattern subgroups of the unipotent upper triangular matrix group. J Algebr Comb 35, 61–92 (2012). https://doi.org/10.1007/s10801-011-0293-5
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DOI: https://doi.org/10.1007/s10801-011-0293-5