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 $Sy{m}^{n}X$ 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 ${\mu }_{Sy{m}^{n}X}\left(t,u,v\right)$. These formulas are then applied to the case of linear algebraic groups.

This new version simplifies the technical aspects in the previous one, and adds several examples, such as complex tori and real Lie groups