On a q-Identity Arising from the Dimension of a Representation of GL(n) over a Finite Field

TL;DR

This paper proves a q-identity related to the dimension of a specific representation of GL(n) over a finite field, simplifying its calculation and aiding in its characterization.

Contribution

It introduces a new q-identity that simplifies the dimension computation of a representation of GL(n) over finite fields, providing insights into its structure.

Findings

Proved a new q-identity for representation dimensions

Simplified the calculation of the representation's dimension

Enabled a detailed description of the representation

Abstract

The present paper proves a $q$-identity, which arises from a representation $\pi_{N,\psi}$ of $\text{GL}_n(\mathbb{F}_q)$. This identity gives a significant simplification for the dimension of $\pi_{N,\psi}$, which allowed the second author to obtain a description of the representation.