Hodge-Deligne Polynomials of Symmetric Products of Algebraic Groups

TL;DR

This paper investigates the mixed Hodge structures of symmetric products of complex algebraic varieties, providing explicit formulas for their Hodge polynomials, especially for linear algebraic groups, based on the cohomology of the original variety.

Contribution

It derives explicit formulas for the mixed Hodge polynomials of symmetric products of algebraic varieties with specific cohomological properties, extending to linear algebraic groups.

Findings

Derived formulas for equivariant mixed Hodge polynomials of symmetric products.

Provided explicit Hodge polynomial formulas for symmetric products of linear algebraic groups.

Enhanced understanding of the Hodge structures in symmetric products of algebraic varieties.

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.