# E-polynomials and geometry of character varieties

With $G=GL(n,\mathbb{C})$, 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 Hodge-Euler (also called $E$-) polynomials of $\mathcal{X}_{\Gamma}G$ and the one for $\mathcal{X}_{\Gamma}^{irr}G$, generalizing a formula of Mozgovoy-Reineke. The proof uses a natural stratification of $\mathcal{X}_{\Gamma}G$ coming from affine GIT and the combinatorics of partitions. Combining our methods with arithmetic ones yields explicit expressions for the $E$-polynomials of all polystable strata of some $GL(n,\mathbb{C})$-character varieties of several groups $\Gamma$, for low values of $n$. For the case $\Gamma=F_{r}$, the free group of rank $r$, using geometric methods and the language of partitions, we prove that $E(\mathcal{X}_{r}SL_{n})=E(\mathcal{X}_{r}PGL_{n})$, for any $n,r\in\mathbb{N}$, settling a conjecture of Lawton-Muñoz. Using this relation, additional explicit computations of polynomials are also provided.

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Alfonso Zamora