Iterative character constructions for algebra groups

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 $n\le 12$. 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.

22 pages

References in corpus (6)

• Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras
• Superinduction for pattern groups
• A Supercharacter Analogue for Normality
• Combinatorial methods of character enumeration for the unitriangular group
• Exotic characters of unitriangular matrix groups
• Base change maps for unipotent algebra groups

Cited by in corpus (4)

• Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras
• Combinatorial methods of character enumeration for the unitriangular group
• Exotic characters of unitriangular matrix groups
• Heisenberg characters, unitriangular groups, and Fibonacci numbers