Orthogonal Determinants of $\mathrm{GL}_n(q)$

TL;DR

This paper derives explicit formulas for orthogonal determinants of certain characters of the general linear group over finite fields, confirming a specific case of Parker’s conjecture and revealing their oddness property.

Contribution

It provides explicit formulas for orthogonal determinants of even-degree orthogonal characters of $ ext{GL}_n(q)$ and verifies their oddness, advancing understanding of character determinants.

Findings

Explicit formulas for orthogonal determinants of $ ext{GL}_n(q)$ characters.

Confirmation that these determinants are 'odd'.

Supports a case of Richard Parker's conjecture.

Abstract

Let $n$ be a positive integer and $q$ be a power of an odd prime. We provide explicit formulas for calculating the orthogonal determinants $\det(\chi)$, where $\chi \in \mathrm{Irr}(\mathrm{GL}_n(q))$ is an orthogonal character of even degree. Moreover, we show that $\det(\chi)$ is "odd". This confirms a special case of a conjecture by Richard Parker.