Caltech/USC Joint Algebra and Geometry Seminar

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 $X^n \times^{S_n} GL_n/T_n$. Our main result establishes that a natural map $\sigma:S_n(X) \to C_n(X)$ 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 also provide a Hermitian variant of our main result. This is joint work with Asvin G, Yifeng Huang, Ruofan Jiang.