Hodge-Deligne Polynomials of Symmetric Products of Algebraic Groups

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