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

Abstract

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 Fq, and show that the number coincides with the virtual Hodge polynomial evaluated at q.

The moduli of representations of degree 2

“…In this case, k( x) = k(x) for the unique point x lying over x. Note that q : C(m) → C(m) induces a bijection of sets C(m)(K) ∼ = C(m)(K) if K is an algebraically closed field of characteristic 2 and that q induces a purely inseparable extension of the function fields of degree 2 (see also [13,Remark 3.3]). Remark 7.35.…”

“…Similarly, the virtual Hodge polynomials of Rep 2 (Γ) air and Ch 2 (Γ) air over C can be calculated from those of Rep 2 (Γ) and the others Rep 2 (Γ) * over C. The existence of such geometric objects as the moduli of representations with several molds helps us to understand relations between the numbers of equivalence classes of representations of Γ over F q and virtual Hodge polynomials of the moduli (cf. [13]).…”

“…In [13], the authors deal with the case that Γ is the free monoid Υ m of rank m. Since Rep 2 (Υ m ) is isomorphic to M 2 × · · · × M 2 (m times), the numbers of the F q -rational points of Rep 2 (Υ m ) air and Ch 2 (Υ m ) air have been calculated explicitly. (The author needs to mention that our strategy to calculate the numbers of the F q -rational points is essentially same as [1] and [7].…”

“…Although his name does not appear in the list of the authors, his contribution to this paper is not ignorable. This article has been inspired by his descriptions of the moduli of representations and approaches from viewpoints of algebraic topology, and so on, which will be written in [13].…”