Counting using Hall Algebras II. Extensions from Quivers

We count the $\mathbb{F}_q$-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 $A_2$ 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.Comment: 18 pages. V2. A missing diagram added. V3. Final version to appear Algebr. Represent. Theory (2015

Cluster Algebras and Semi-invariant Rings I. Triple Flags

We prove that each semi-invariant ring of the complete triple flag of length $n$ is an upper cluster algebra associated to an ice hive quiver. We find a rational polyhedral cone ${\sf G}_n$ such that the generic cluster character maps its lattice points onto a basis of the cluster algebra. As an application, we use the cluster algebra structure to find a special minimal set of generators for these semi-invariant rings when $n$ is small.Comment: Text overlap with arXiv:1210.1888 by other authors. v2. notation revised, typos corrected. v3. a shortened version. remove a wrong citation on local acyclicity, simplify the proof of Lemma 8.