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

Abstract

With G=GL(n,C), let ${\mathcal{X}}_{\Gamma }G$ be the G-character variety of a given finitely presented group $\Gamma$, and let ${\mathcal{X}}_{\Gamma }^{irr}G\subset {\mathcal{X}}_{\Gamma }G$ 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 ${\mathcal{X}}_{\Gamma }G$ and the one for ${\mathcal{X}}_{\Gamma }^{irr}G$, generalizing a formula of Mozgovoy-Reineke [MR]. The proof uses a natural stratification of ${\mathcal{X}}_{\Gamma }G$ 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 of the irreducible stratum of GL(n,C)-character varieties of some groups $\Gamma$, including surface groups, free groups, and torus knot groups, for low values of $n$.

24 pages, The free group case moved to another article; application to Cartan brane included; a couple references added

References in corpus (8)

• Counting Gauge Invariants: the Plethystic Program
• Moduli of representations of quivers
• A computational criterion for the Kac conjecture
• Serre polynomials of $S{L}_n$- and $PG{L}_n$-character varieties of free groups
• Branes on the singular locus of the Hitchin system via Borel and other parabolic subgroups
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Cited by in corpus (2)

• Serre polynomials of $S{L}_n$- and $PG{L}_n$-character varieties of free groups
• Plethystic exponential calculus and characteristic polynomials of permutations