Counting Using Hall Algebras Ii. Extensions from Quivers

We count the Fq-rational points of GIT quotients of quiver representations with relations. We focus on two types of algebras – one is one-point extended from a quiver Q, and the other is the Dynkin A2 tensored with Q. For both, we obtain explicit formulas. We study when they are polynomial-count. We follow the similar line as in the first paper but algebraic manipulations in Hall algebra will be replaced by corresponding geometric constructions. Introduction We continue our development on algorithms to count the points of various representation varieties of a quiver with relations. In [5], we applied serval counting characters to the Harder-Narasimhan identity (2.1) in the Hall algebra of a quiver and obtained several interesting formulas. All characters that we considered are originated from Reineke’s counting character ∫ from the Hall algebra to certain quantum power series ring. Unfortunately ∫ fails to be an algebra morphism for non-hereditary algebras, though Harder-Narasimhan identity exists quite generally. However, applying the same map ∫ to the HN-identity can still generate effective counting formulas. We will follow the similar line as the first paper. The only change is that we replace algebraic manipulations in the Hall algebras by corresponding geometric constructions. We first state the main results of this notes. Let k to be the finite field Fq with q elements and A be any basic algebra presented by A = kQ/I. Fix a slope function μ, and we denote by Repα(A) the variety of α-dimensional μ-semistable representations of A, and by Modα(A) its GIT quotient. Lemma 0.1. |Repα(A)| = ∑ (−1)|Frepα1···αs(A)|, where the sum runs over all decomposition α1 + · · · + αs = α of α into non-zero dimension vectors such that μ( ∑k l=1 αl) < μ(α) for k < s. We will define the key varieties Frepα1···αs(A) in Section 1. In particular, if all Frep varieties can be effectively counted, then so are Repα(A). The map ∫ have so-called ∆ and S analogs. They are defined in [5] as ∫ ∆ and S ∫ . Here, ∆ and S are related to the comultiplication and the antipode in the Hall algebra. In our geometric setting, Lemma 0.1 and Frep varieties have ∆ and S analogs as well. Recall that a variety X is called polynomial-count (or has a counting polynomial) if there exists a (necessarily unique) polynomial PX = ∑ aiti ∈ C[t] such that 2010 Mathematics Subject Classification. Primary 16G10; Secondary 14D20,14N10.