Stiefel Whitney Classes for Real representations of $\mathrm{GL}_2(\mathbb{F}_q)$

TL;DR

This paper computes the total Stiefel Whitney class for real representations of the group GL_2 over finite fields of odd order, providing explicit formulas for the obstruction class based on character values.

Contribution

It introduces a method to explicitly calculate the obstruction class of real representations of GL_2(F_q) using character theory, extending understanding of their topological invariants.

Findings

Explicit formulas for the total Stiefel Whitney class of representations.

Character-based expression for the obstruction class when determinant is 1.

Enhanced understanding of topological invariants of finite group representations.

Abstract

We compute the total Stiefel Whitney class for a real representation $\pi$ of $\mathrm{GL}_2(\mathbb{F}_q)$, where $q$ is odd. The obstruction class of $\pi$ is defined to be the Stiefel Whitney class of lowest positive degree that does not vanish. We provide an expression for the obstruction class of $\pi$ in terms of its character values if $\det\pi=1$.