Counting characters of small degree in upper unitriangular groups

AbstractLet Un denote the group of upper n×n unitriangular matrices over a fixed finite field F of order q. That is, Un consists of upper triangular n×n matrices having every diagonal entry equal to 1. It is known that the degrees of all irreducible complex characters of Un are powers of q. It was conjectured by Lehrer that the number of irreducible characters of Un of degree qe is an integer polynomial in q depending only on e and n. We show that there exist recursive (for n) formulas that this number satisfies when e is one of 1,2 and 3, and thus show that the conjecture is true in those cases