Total Stiefel Whitney classes for real representations of $\mathrm{GL}_n$ over $\mathbb{F}_q, \mathbb{R}$ and $\mathbb{C}$

TL;DR

This paper computes the total Stiefel Whitney classes for real representations of general linear groups over finite fields, real numbers, and complex numbers, linking these classes to character values on specific elements.

Contribution

It provides explicit formulas for total Stiefel Whitney classes of real representations of $ ext{GL}_n$ over various fields, connecting topological invariants to character theory.

Findings

Formulas for $ ext{GL}_n( ext{finite field})$ with odd $q$

Total Stiefel Whitney classes for $ ext{GL}_n( ext{real})$

Total Stiefel Whitney classes for $ ext{GL}_n( ext{complex})$

Abstract

We compute the total Stiefel Whitney class for a real representation $\pi$ of $\mathrm{GL}_n(\mathbb{F}_q)$, where $q$ is odd in terms of character values of $\pi$ on order $2$ diagonal elements. We also compute the total Stiefel Whitney class of real representations of $\mathrm{GL}_n(\mathbb{R})$ and $\mathrm{GL}_n(\mathbb{C})$.