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

We study the asymptotic behavior of Stiefel–Whitney classes of irreducible orthogonal representations of the finite general linear groups ${\mathrm{G}\mathrm{L}}_n({\mathbb{F}}_q)$. 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 $n\to \mathrm{\infty }$, 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 $q\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$, 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 $q\to \mathrm{\infty }$, the behavior is markedly different. Focusing on ${\mathrm{G}\mathrm{L}}_2({\mathbb{F}}_q)$, we show that the second Stiefel–Whitney class vanishes with limiting probability $3\mathrm{/}8$ among irreducible orthogonal representations.