Carlos Florentino, Azizeh Nozad, Alfonso Zamora
Character varieties and Higgs bundles
With G=GL(n,C), let XΓG be the G-character variety of a given finitely presented group Γ, and let XirrΓG⊂XΓG 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 XΓG and the one for XirrΓG, generalizing a formula of Mozgovoy-Reineke [MR]. The proof uses a natural stratification of XΓG 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 Γ, including surface groups, free groups, and torus knot groups, for low values of n.
2019
Florentino, Carlos, Azizeh Nozad, and Alfonso Zamora. “Generating series for the E-polynomials of GL(n,C)-character varieties” arXiv preprint arXiv:1902.06837 (2019). To appear on Math Nachrichten