Asymptotic Vanishing of Stiefel--Whitney Classes for $\mathrm{GL}_n(\mathbb{F}_q)$

TL;DR

This paper investigates the asymptotic behavior of Stiefel--Whitney classes of irreducible orthogonal representations of finite general linear groups, revealing that most become trivial as rank increases, with specific results for small ranks.

Contribution

It establishes the asymptotic vanishing of certain Stiefel--Whitney classes for large rank groups and analyzes their behavior in fixed rank limits, providing new insights into the topology of these representations.

Findings

As rank n increases, the proportion of representations with trivial first and second Stiefel--Whitney classes approaches 1.

For q ≡ 1 mod 4, the fourth Stiefel--Whitney class also tends to vanish in the large rank limit.

In fixed rank and q → ∞, the second Stiefel--Whitney class vanishes with probability 5/16 for GL_2(F_q).

Abstract

We study the asymptotic behavior of Stiefel--Whitney classes of irreducible orthogonal representations of the finite general linear groups $\mathrm{GL}_n(\mathbb{F}_q)$. Building on recent formulas expressing these classes in terms of character values at elements of order dividing $2$, we relate questions about characteristic classes to problems of $2$-adic divisibility of character values. For fixed odd $q$, we show that as $n \to \infty$, the values of irreducible orthogonal characters become highly divisible by powers of $2$ for almost all representations. As a consequence, the proportion of irreducible orthogonal representations with trivial first and second Stiefel--Whitney classes tends to $1$, and if $q \equiv 1 \pmod{4}$, the same holds for the fourth Stiefel--Whitney class. In particular, almost all orthogonal representations are spinorial in the large rank limit. In contrast, when the rank is fixed and $q \to \infty$, the behavior is markedly different. Focusing on $\mathrm{GL}_2(\mathbb{F}_q)$, we show that the second Stiefel--Whitney class vanishes with limiting probability $5/16$ among irreducible orthogonal representations.