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In this paper we study the topology of the moduli spaces of representationsof degree $2$ for free monoids. We calculate the virtual Hodge polynomials ofthe 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 showthat 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