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 Fq, and show that the number coincides with the virtual Hodge polynomial evaluated at q.
"Virtual Hodge polynomials of the moduli spaces of representations of degree 2 for free monoids." Kodai Math. J. 39 (1) 80 - 109, March 2016. https://doi.org/10.2996/kmj/1458651693