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

TL;DR

This paper computes the virtual Hodge polynomials of moduli spaces of 2-dimensional representations of free monoids and confirms their correspondence with counts over finite fields.

Contribution

It provides explicit calculations of virtual Hodge polynomials for various representation types and links these to finite field point counts.

Findings

Virtual Hodge polynomials computed for several representation types

Number of isomorphism classes over finite fields matches polynomial evaluations

Establishes a correspondence between topological invariants and finite field counts

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 ${\Bbb F}_q$, and show that the number coincides with the virtual Hodge polynomial evaluated at $q$.