Shell Tableaux: A set partition analogue of vacillating tableaux

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, and develop an analog of Young tableaux, known as shell tableaux, to index paths in this graph.