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

Abstract

With 𝐺 = 𝐺𝐿(𝑛, ℂ), let  Γ 𝐺 be the 𝐺-character variety of a given finitely presented group Γ, and let  𝑖𝑟𝑟 Γ 𝐺 ⊂  Γ 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  Γ 𝐺 and the one for  𝑖𝑟𝑟 Γ 𝐺, generalizing a formula of Mozgovoy-Reineke. The proof uses a natural stratification of  Γ 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)-character varieties of some groups Γ, including surface groups, free groups, and torus knot groups, for low values of 𝑛. K E Y W O R D Scharacter varieties, E-polynomials, Hodge theory, representations of finitely presented groups INTRODUCTIONLet 𝐺 be a complex reductive algebraic group, Γ be a finitely presented group, such as the fundamental group of a compact manifold or a finite 𝐶𝑊-complex, and letThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.