Iterative character constructions for algebra groups

TL;DR

This paper introduces a new family of orthogonal characters for algebra groups that decompose supercharacters, providing formulas, irreducibility conditions, and applications to the unitriangular group, enhancing understanding of character structures.

Contribution

It constructs a novel family of characters for algebra groups, generalizes existing theorems, and connects to recent computations on irreducible characters of unitriangular groups.

Findings

Characters decompose supercharacters into nonnegative integer combinations.

Derived explicit formulas and irreducibility conditions for these characters.

Showed that all irreducible characters of UT_n(q) are Kirillov functions if and only if n ≤ 12.

Abstract

We construct a family of orthogonal characters of an algebra group which decompose the supercharacters defined by Diaconis and Isaacs. Like supercharacters, these characters are given by nonnegative integer linear combinations of Kirillov functions and are induced from linear supercharacters of certain algebra subgroups. We derive a formula for these characters and give a condition for their irreducibility; generalizing a theorem of Otto, we also show that each such character has the same number of Kirillov functions and irreducible characters as constituents. In proving these results, we observe as an application how a recent computation by Evseev implies that every irreducible character of the unitriangular group $\UT_n(q)$ of unipotent $n\times n$ upper triangular matrices over a finite field with $q$ elements is a Kirillov function if and only if $n\leq 12$. As a further application, we discuss some more general conditions showing that Kirillov functions are characters, and describe some results related to counting the irreducible constituents of supercharacters.