Asymptotic Vanishing of Stiefel--Whitney Classes for $\mathrm{GL}_n(\mathbb{F}_q)$
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| Format: | Preprint |
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2026
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| _version_ | 1866913159093157888 |
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| author | Ray, Anwesh |
| author_facet | Ray, Anwesh |
| contents | 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 $3/8$ among irreducible orthogonal representations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_27235 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic Vanishing of Stiefel--Whitney Classes for $\mathrm{GL}_n(\mathbb{F}_q)$ Ray, Anwesh Representation Theory Algebraic Topology Group Theory Number Theory 20G05, 20G40, 20J05 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 $3/8$ among irreducible orthogonal representations. |
| title | Asymptotic Vanishing of Stiefel--Whitney Classes for $\mathrm{GL}_n(\mathbb{F}_q)$ |
| topic | Representation Theory Algebraic Topology Group Theory Number Theory 20G05, 20G40, 20J05 |
| url | https://arxiv.org/abs/2604.27235 |