Hodge-Deligne polynomials of character varieties of abelian groups

TL;DR

This paper computes Hodge-Deligne polynomials for character varieties of free abelian groups, providing explicit formulas especially for G=GL(n,C) and SL(n,C) using combinatorial methods.

Contribution

It introduces explicit formulas for mixed Hodge-Deligne polynomials of character varieties of abelian groups, extending previous knowledge with new combinatorial expressions.

Findings

Explicit formulas for Hodge-Deligne polynomials of character varieties.

Derived combinatorial expressions for G=GL(n,C) and SL(n,C).

Applicable to quotients of varieties with simple mixed Hodge structures.

Abstract

Let F be a finite group and X be a complex quasi-projective F-variety. For r in N, we consider the mixed Hodge-Deligne polynomials of quotients X^r/F, where F acts diagonally, and compute them for certain classes of varieties X with simple mixed Hodge structures. A particularly interesting case is when X is the maximal torus of an affine reductive group G, and F is its Weyl group. As an application, we obtain explicit formulae for the Hodge-Deligne and E-polynomials of (the distinguished component of) G-character varieties of free abelian groups. In the cases G=GL(n,C) and SL(n,C) we get even more concrete expressions for these polynomials, using the combinatorics of partitions.