A conjecture about characters of finite pattern groups

Abstract

Inspired by Kirillov's orbital method, we propose a conjecture (Conjecture 1.7), which suggests an explicit and exhaustive construction of characters for finite pattern groups via coadjoint orbits. First, we establish the construction for irreducible characters of degree q for finite pattern groups. Second, we verify the conjecture for the case of finite pattern groups contained in U4(Fq), where Un stands for the full unitriangular group of rank n. Finally, we classify irreducible characters for Gn(Fq), a generalization of Heisenberg group, where Gn is the pattern subgroup of Un, associated to the closed set{(1,m),(k,n),(i,n+1−i)|1<m,k<n,1⩽i⩽[n2]},3⩽n∈N. Hence Conjecture 1.7 and the analogous conjectures of Higman, Lehrer and Isaacs holds for Gn(Fq).