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

Abstract

Let G be a complex reductive group and XrG denote the G-character variety of the free group of rank r. Using geometric methods, we prove that E(XrSLn) = E(XrP GLn), for any n, r ∈ N, where E(X) denotes the Serre (also known as E-) polynomial of the complex quasi-projective variety X, settling a conjecture of Lawton-Muñoz 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 XrSLn and XrP GLn. 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 GLn-character varieties over finite fields.

On character varieties of singular manifolds

“…This is due to the fact that, for the stratum X(Σ g,b ) , the set of feasible eigenvalues and antidiagonal elements are required to lie in a hyperplane. However, this equation vanishes for g = 0 but, in that case, we only need to consider the quotient of a free group, as studied in Section 7.1 of [14] or [11].…”

“…These are the so-called knot character varieties and have been studied, for instance, in [17,34,35] for torus knots or in [22] for the figure eight knot. In addition, in [11] the E-polynomial is computed for the complement of unknoted links (i.e. free fundamental group), and in [12] for tori X = S 1 × .…”

Representation Varieties of Twisted Hopf Links

“…The most studied case is the character variety of trivial links, i.e. representations of the free group, addressed in works such as [6,7,18] (focused on the topology) and [1,8,19] (computing the E-polynomials). Very recently, more complicated links were studied, such as the twisted Alexander polynomial for the Borromean link in [2].…”