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.

Date and Venue

Start Date
Venue
Room 1.08
End Date

Speaker

Alfonso Zamora

Speaker's Institution

Universidad CEU San Pablo, Madrid, Spain

Files

Area

Geometry and Topology

Financiamento