Counting irreducible representations of large degree of the upper triangular groups

TL;DR

This paper constructs large degree irreducible representations of the upper triangular group over finite fields and determines the counts of the top three largest degree irreducible representations.

Contribution

It provides explicit constructions and counts of the largest irreducible representations of upper triangular groups for n ≥ 7.

Findings

Constructed large degree irreducible representations for U_n(q)

Determined the number of irreducible representations of top three degrees

Focused on groups with n ≥ 7

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.