Branching rules in the ring of superclass functions of unipotent upper-triangular matrices

TL;DR

This paper explores the combinatorial and algebraic structures of supercharacter theory for unipotent upper-triangular matrices, revealing connections to symmetric functions and representation theory.

Contribution

It introduces a link between supercharacter theory and non-commutative symmetric functions, and analyzes structure constants for supercharacter operations.

Findings

Connection to non-commutative symmetric functions established

Structure constants for supercharacter operations computed

Enhanced understanding of supercharacter theory's combinatorial structure

Abstract

It is becoming increasingly clear that the supercharacter theory of the finite group of unipotent upper-triangular matrices has a rich combinatorial structure built on set-partitions that is analogous to the partition combinatorics of the classical representation theory of the symmetric group. This paper begins by exploring a connection to the ring of symmetric functions in non-commuting variables that mirrors the symmetric group's relationship with the ring of symmetric functions. It then also investigates some of the representation theoretic structure constants arising from the restriction, tensor products and superinduction of supercharacters in this context.