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.
From: Yifan Wei [view email]
[v1]
Wed, 15 Oct 2025 10:21:35 UTC (25 KB)
[v2]
Mon, 27 Oct 2025 06:24:41 UTC (26 KB)
[v3]
Mon, 15 Jun 2026 06:30:38 UTC (26 KB)
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