On the E-polynomials of a family of Character Varieties

TL;DR

This paper calculates the E-polynomials of certain twisted character varieties by establishing polynomial countability and using character theory, Frobenius formulas, and combinatorial tools to derive explicit formulas and Euler characteristics.

Contribution

It introduces a method to compute E-polynomials of a family of character varieties using character tables and combinatorial techniques, providing explicit formulas and Euler characteristics.

Findings

E-polynomials are explicitly computed for the family of varieties.

The varieties are shown to have polynomial point counts over finite fields.

Euler characteristics are derived from the E-polynomials.

Abstract

We compute the E-polynomials of a family of twisted character varieties by proving they have polynomial count, and applying a result of N. Katz on the counting functions. To compute the number of GF(q)-points of these varieties as a function of q, we used a formula of Frobenius. Our calculations made use of the character tables of Gl(n,q) and Sl(n,q), previously computed by J. A. Green and G. Lehrer, and a result of Hanlon on the M\"obius function of a subposet of set-partitions. The Euler Characteristics of these character varieties are calculated with these polynomial.