Coadjoint orbits of Row Closed Subgroups of $U_n(q)$

Abstract

Let $n$ be a natural number, let $q$ be a prime power, and let $U_n(q)$ denote the group of unitriangular $n\times n$ matrices over the finite field $\mathbb{F}_q$ with $q$ elements. Row closed and column closed subgroups $U$ of $U_n(q)$ are special pattern subgroups obtained by deleting entire rows or, respectively, entire columns (apart from the diagonal entries). The supercharacters of the Andr\'{e}-Yan supercharacter theory of $U$ are afforded by the orbit modules arising from a monomial action of $U$ on the character group of the Lie algebra of $U$. We classify the corresponding orbits for row closed subgroups $U$, that is, we determine all orbits and identify which of the associated orbit modules are isomorphic and which afford orthogonal supercharacters. The classification for column closed subgroups follows from the mirror map, which reflects matrices across the antidiagonal.