Generating series for the E-polynomials of GL(n,C)$GL(n,{\mathbb {C}})$-character varieties

With G=GL(n,C)$G=GL(n,\mathbb {C})$, let X Gamma G$\mathcal {X}_{\Gamma }G$ be the G-character variety of a given finitely presented group Gamma, and let X Gamma irrG subset of X Gamma G$\mathcal {X}_{\Gamma }<^>{irr}G\subset \mathcal {X}_{\Gamma }G$ be the locus of irreducible representation conjugacy classes. We provide a concrete relation, in terms of plethystic functions, between the generating series for E-polynomials of X Gamma G$\mathcal {X}_{\Gamma }G$ and the one for X Gamma irrG$\mathcal {X}_{\Gamma }<^>{irr}G$, generalizing a formula of Mozgovoy-Reineke. The proof uses a natural stratification of X Gamma G$\mathcal {X}_{\Gamma }G$ coming from affine GIT, the combinatorics of partitions, and the formula of MacDonald-Cheah for symmetric products; we also adapt it to the so-called Cartan brane in the moduli space of Higgs bundles. Combining our methods with arithmetic ones yields explicit expressions for the E-polynomials, and Euler characteristics, of the irreducible stratum of GL(n,C)$GL(n,\mathbb {C})$-character varieties of some groups Gamma, including surface groups, free groups, and torus knot groups, for low values of n.

Automated summary

In this paper, we generalize a formula of Mozgovoy-Reineke by providing a concrete relation, in terms of plethystic functions, between the generating series for E-polynomials of X_Gamma_G and X_Gamma_irrG. We prove this relation using a natural stratification of X_Gamma_G, combinatorics of partitions, and the formula of MacDonald-Cheah for symmetric products. We also adapt our method to the Cartan brane in the moduli space of Higgs bundles. Furthermore, combining our methods with arithmetic ones allows us to obtain explicit expressions for the E-polynomials and Euler characteristics of the irreducible stratum of GL(n,C)-character varieties of some groups Gamma, for low values of n.