Virtual Hodge polynomials of the moduli spaces of representations of degree 2 for free monoids

Abstract

In this paper we study the topology of the moduli spaces of representations of degree 2 for free monoids. We calculate the virtual Hodge polynomials of the character varieties for several types of 2-dimensional representations. Furthermore, we count the number of isomorphism classes for each type of 2-dimensional representations over any finite field F_q, and show that the number coincides with the virtual Hodge polynomial evaluated at q.