Matrix Points on Varieties

TL;DR

The paper establishes a cohomological equivalence between the Bosonic moduli $C_n(X)$ of commuting $n\times n$ matrices and a Fermionic partner $S_n(X)$, via a natural map $\sigma$, under characteristic zero or local-homogeneity assumptions. It provides explicit Betti-number formulas and a Macdonald-type generating series for $C_n(X)$ and uses symmetric-function techniques to express cohomology in terms of graded $S_n$-characters, leading to product formulas tied to Betti zeta functions. A Hermitian variant over $\mathbb{R}$ is developed, and strong equivariant formality results are derived, yielding a precise cohomological model consistent with Weil-restriction heuristics. The framework recovers classical commuting varieties as special cases and delivers a comprehensive combinatorial toolkit (Frobenius characters, principal specializations, and generating series) to compute cohomology and Betti numbers for these non-commutative moduli spaces. Overall, the work bridges non-commutative Weil restrictions, Boson-Fermion duality, and explicit cohomological calculations for matrix-point moduli spaces.

Abstract

We study the cohomology of $C_n(X)$, the moduli space of commuting $n$-by-$n$ matrices satisfying the equations defining a variety $X$. 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 ``Fermionic" counterpart $S_n(X)$, constructed as a convolution $X^n \times^{S_n} \mathrm{GL}_n/\mathrm{T}_n$. Our main result establishes that a natural map $σ\colon S_n(X) \to C_n(X)$ induces an isomorphism on $\ell$-adic cohomology under mild conditions on $X$ or the characteristic of the field. This confirms a heuristic derived from the classical theory of Weil restrictions and highlights a version of Boson-Fermion correspondence. Furthermore, we derive explicit combinatorial formulae for the Betti numbers of $C_n(X)$ and a Macdonald-type generating series. Finally we prove a Hermitian variant of our main result.

Theorems & Definitions (53)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1
• Lemma 1
• Lemma 2
• proof
• Proposition 1: Stalk of $Rp_{1*}\underline{\mathbb{Q}}_\ell$
• proof