Counting irreducible representations of large degree of the upper triangular groups

Abstract

Let $U_n(q)$ be the upper triangular group of degree $n$ over the finite field $\F_q$ with $q$ elements. In this paper, we present constructions of large degree ordinary irreducible representations of $U_n(q)$ where $n\geq 7$, and then determine the number of irreducible representations of largest, second largest and third largest degrees.