Restrictions of rainbow supercharacters

TL;DR

This paper introduces a new approach to understanding the restriction of rainbow supercharacters in unipotent upper-triangular matrix groups, using intermediate modules and a novel q-analogue of binomial coefficients.

Contribution

It develops a method to factor structure constants in supercharacter restrictions and solves the problem completely for rainbow supercharacters, introducing new combinatorial tools.

Findings

Complete solution for rainbow supercharacter restrictions

Introduction of a q-analogue of binomial coefficients depending on a poset

Development of intermediate modules to facilitate restriction computations

Abstract

The maximal subgroup of unipotent upper-triangular matrices of the finite general linear groups are a fundamental family of $p$-groups. Their representation theory is well-known to be wild, but there is a standard supercharacter theory, replacing irreducible representations by super-representations, that gives us some control over its representation theory. While this theory has a beautiful underlying combinatorics built on set partitions, the structure constants of restricted super-representations remain mysterious. This paper proposes a new approach to solving the restriction problem by constructing natural intermediate modules that help "factor" the computation of the structure constants. We illustrate the technique by solving the problem completely in the case of rainbow supercharacters (and some generalizations). Along the way we introduce a new $q$-analogue of the binomial coefficients that depend on an underlying poset.