Hodge-Deligne Polynomials of Symmetric Products of Algebraic Groups

Abstract

Let $X$ be a complex quasi-projective algebraic variety. In this paper we study the mixed Hodge structures of the symmetric products $Sym^{n}X$ when the cohomology of $X$ is given by exterior products of cohomology classes with odd degree. We obtain an expression for the equivariant mixed Hodge polynomials $\mu_{X^{n}}^{S_{n}}\left(t,u,v\right)$, codifying the permutation action of $S_{n}$ as well as its subgroups. This allows us to deduce formulas for the mixed Hodge polynomials of its symmetric products $\mu_{Sym^{n}X}\left(t,u,v\right)$. These formulas are then applied to the case of linear algebraic groups.