Megan Ly and Shawn Burkett (CU) Schur--Weyl Duality for Unipotent Upper Triangular Matrices AND Lattices of normal subgroups and supercharacter theory
Tue, Oct. 31 3pm (MATH 350)
Topology
Katharine Adamyk TBA
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Emily Gari, the Gemmill Engineering, Math & Physics Library’s Math Liaison Librarian, will provide an overview of library resources & services and literature review techniques, and answer any and all library related questions.
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Schur--Weyl duality is a fundamental framework in combinatorial representation theory. It intimately relates the irreducible characters of the symmetric group to the irreducible characters of the general linear group via their commuting actions on tensor space. We investigate the analog of Schur-Weyl duality for the group of unipotent upper triangular matrices over a nite eld. In this case, the character theory of these upper triangular matrices is unattainable. Thus we employ a generalization, known as supercharacter theory, to create a striking variation on the character theory of the symmetric group with combinatorics based on set partitions. We present a combinatorial structure that encodes the decomposition of a tensor space into supercharacters in order to describe the maps that centralize the action of the group of unipotent upper triangular matrices.
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A supercharacter theory of a nite group is an approximation of its character theory where the role of the irreducible characters is played by a certain set of characters, called supercharacters, which enjoy some similar algebraic properties. Given a supercharacter theory of $G$, a lattice of normal subgroups of $G$ can be associated via the kernels of the supercharacters. Conversely, given any lattice $\mathcal{L}$ of normal subgroups of $G$, a supercharacter theory of $G$ can be constructed whose associated lattice is $\mathcal{L}$ and which is as coarse as possible, in some sense. In this talk, we will discuss some properties of these lattices, as well as the possibility of constructing finer supercharacter theories from a lattice of normal subgroups by specifying supercharacter theories on each covering relation.