Serre polynomials of $SL_n$- and $PGL_n$-character varieties of free groups

TL;DR

This paper proves that the Serre polynomials of $SL_n$- and $PGL_n$-character varieties of free groups are equal, confirming a conjecture, and provides explicit calculations of these polynomials and Euler characteristics.

Contribution

It establishes the equality of Serre polynomials for $SL_n$ and $PGL_n$ character varieties, using geometric stratification methods, and computes these polynomials explicitly.

Findings

Proved $E( ext{character variety of } SL_n) = E( ext{character variety of } PGL_n)$.

Computed explicit Serre polynomials and Euler characteristics for these varieties.

Validated the conjecture of Lawton-Mu ilde{n}oz for all ranks and dimensions.

Abstract

Let $G$ be a complex reductive group and $\mathcal{X}_{r}G$ denote the $G$-character variety of the free group of rank $r$. Using geometric methods, we prove that $E(\mathcal{X}_{r}SL_{n})=E(\mathcal{X}_{r}PGL_{n})$, for any $n,r\in\mathbb{N}$, where E(X) denotes the Serre (also known as E-) polynomial of the complex quasi-projective variety $X$, settling a conjecture of Lawton-Mu\~noz in [LM]. The proof involves the stratification by polystable type introduced in [FNZ], and shows moreover that the equality of E-polynomials holds for every stratum and, in particular, for the irreducible stratum of $\mathcal{X}_{r}SL_{n}$ and $\mathcal{X}_{r}PGL_{n}$. We also present explicit computations of these polynomials, and of the corresponding Euler characteristics, based on our previous results and on formulas of Mozgovoy-Reineke for $GL_{n}$-character varieties over finite fields.