On monomial linearisation and supercharacters of pattern subgroups

TL;DR

This paper develops a general framework for supercharacters of pattern subgroups of upper unitriangular groups, extending previous constructions and classifying the resulting supercharacters through orbit analysis.

Contribution

It generalizes Yan's coadjoint cluster representations to column closed pattern subgroups and provides a complete classification of supercharacters for these groups.

Findings

Constructed supercharacters via monomial linearisation for pattern subgroups.

Classified supercharacters by describing orbits and Hom-spaces.

Extended supercharacter theory to a broader class of subgroups.

Abstract

Column closed pattern subgroups $U$ of the finite upper unitriangular groups $U_n(q)$ are defined as sets of matrices in $U_n(q)$ having zeros in a prescribed set of columns besides the diagonal ones. We explain Jedlitschky's construction of monomial linearisation and apply this to $C U$ yielding a generalisation of Yan's coadjoint cluster representations. Then we give a complete classification of the resulting supercharacters, by describing the resulting orbits and determining the Hom-spaces between orbit modules.