  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.
