Shell Tableaux: A set partition analogue of vacillating tableaux
Megan Ly

TL;DR
This paper introduces shell tableaux, a combinatorial structure analogous to vacillating tableaux, to study supercharacter theory of unipotent upper triangular matrices, providing formulas for supercharacter restriction and induction.
Contribution
It develops shell tableaux as a new combinatorial tool for analyzing supercharacters of unipotent upper triangular matrices, extending the analogy with vacillating tableaux.
Findings
Provides a combinatorial formula for supercharacter restriction and induction.
Creates a graph encoding tensor space decomposition.
Introduces shell tableaux to index paths in the graph.
Abstract
Schur-Weyl duality is a fundamental framework in combinatorial representation theory. It intimately relates the irreducible representations of a group to the irreducible representations of its centralizer algebra. We investigate the analog of Schur-Weyl duality for the group of unipotent upper triangular matrices over a finite field. In this case, the character theory of these upper triangular matrices is "wild" or unattainable. Thus we employ a generalization, known as supercharacter theory, that creates a striking variation on the character theory of the symmetric group with combinatorics built from set partitions. In this paper, we present a combinatorial formula for calculating a restriction and induction of supercharacters based on statistics of set partitions and seashell inspired diagrams. We use these formulas to create a graph that encodes the decomposition of a tensor space,…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Algebraic structures and combinatorial models · Random Matrices and Applications
