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 XΓG$\mathcal {X}_{\Gamma }G$ be the G‐character variety of a given finitely presented group Γ, and let XΓirrG⊂XΓG$\mathcal {X}_{\Gamma }^{irr}G\subset \mathcal {X}_{\Gamma }G$ 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 XΓG$\mathcal {X}_{\Gamma }G$ and the one for XΓirrG$\mathcal {X}_{\Gamma }^{irr}G$, generalizing a formula of Mozgovoy–Reineke. The proof uses a natural stratification of XΓG$\mathcal {X}_{\Gamma }G$ 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 Γ, including surface groups, free groups, and torus knot groups, for low values of n.