Matrix Points on Varieties

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 $X^n{\times }^{S_n}{\mathrm{G}\mathrm{L}}_n\mathrm{/}{\mathrm{T}}_n$. Our main result establishes that a natural map $\sigma \text{\hspace{0.17em}⁣}:S_n(X)\to C_n(X)$ induces an isomorphism on $\mathrm{\ell }$-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.