On the number of stable quiver representations over finite fields

Abstract

a b s t r a c tWe prove a new formula for the generating function of polynomials counting absolutely stable representations of quivers over finite fields. The case of irreducible representations is studied in more detail.

Counting points of homogeneous varieties over finite fields

“…A similar positivity result has been conjectured by Mozgovoy and Reineke in the setting of quiver moduli (see [MR07,Rem. 6.5] and also [Re08, Conj.…”

Moduli of representations of quivers

“…One of the possible directions towards a study of the global geometry of quiver moduli pursued by the author in [19,46,52,54,55,56,57,58] is the determination of Betti numbers of quiver moduli. We will first consider the question why knowledge of the Betti numbers should be interesting for such a study.…”

“…Bases on this, one defines mutually inverse operators Exp(f ) = exp(Ψ(f )) and Log(f ) = Ψ −1 (log(f )) on F . Using these concepts, an explicit formula for the counting polynomials is proved in [46]:…”

“…Based on computer experiments for the m-loop quiver and dimensions up to 12, the following conjecture is made in [46]:…”

“…Based on this, our third main theme is the discussion of the methods and results of the author's (and others) work [19,46,52,53,54,55,56,57,58]. We focus on the (essentially easy) methods which are used in these papers, namely torus localization in section 7, counting points over finite fields in section 8, Hall algebras in section 9, and framings of moduli in section 10.…”