Matrix Points on Varieties

Asvin G; Yifeng Huang; Ruofan Jiang; Yifan Wei

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arXiv · 2510.13380

Matrix Points on Varieties

Asvin G·Yifeng Huang·Ruofan Jiang·Yifan Wei

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Abstract

We study the cohomology of $C_n(X)$, the moduli space of commuting $n$-by-$n$ matrices satisfying the equations defining a quasi-projective scheme $X$. This space can be viewed as a non-commutative Weil restriction from the algebra of $n$-by-$n$ matrices to the ground field. We introduce a semi-simple counterpart $S_n(X)$, defined as the quotient of $X^n \times \mathrm{GL}_n/\mathrm{T}_n$ by the diagonal $S_n$ action. We show that there exists a natural map $\sigma \colon S_n(X) \to C_n(X)$ inducing isomorphism on $\ell$-adic cohomology under mild restrictions on $X$ or the characteristic of the field. This confirms a heuristic derived from Weil restrictions. Furthermore, we provide explicit combinatorial formulae for the Betti numbers of $C_n(X)$ and prove a Macdonald-type generating series. A version for Hermitian matrix point is also proved in the last section.

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BibTeX RIS

Asvin G, Yifeng Huang, Ruofan Jiang, Yifan Wei. 2025-10-15. Matrix Points on Varieties. https://arxiv.org/abs/2510.13380

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