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 (2008) [6]. 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 (2010) [20], 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 (2010) [7] implies that every irreducible character of the unitriangular group UT n ( q ) of unipotent n × n upper triangular matrices over a finite field with q elements is a Kirillov function if and only if n ⩽ 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.