Let X be a complex quasi-projective algebraic variety. In this paper we study the mixed Hodge structures of the symmetric products \(\mathrm {Sym}^{n}X\) 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 \(\mu _{\mathrm {Sym}^{n}X}\left( t,u,v\right) \). These formulas are then applied to the case of linear algebraic groups.

设X是一个复杂的拟射影代数簇。在本文中，我们研究了当X的上同调由奇次上同调类的外积给出时对称积\(\mathrm {Sym}^{n}X\)的混合 Hodge 结构。我们获得了等变混合霍奇多项式\(\mu _{X^{n}}^{S_{n}}\left( t,u,v\right) \)的表达式，编码了\( S_{n}\)及其子群。这使我们能够推导出其对称乘积\(\mu _{\mathrm {Sym}^{n}X}\left( t,u,v\right) \)的混合霍奇多项式的公式。然后将这些公式应用于线性代数群的情况。