The representations of dimension vector α of the quiver Q can be parametrised by a vector space R(Q,α) on which an algebraic group Gl(α) acts so that the set of orbits is bijective with the set of isomorphism classes of representations of the quiver. We describe the semi–invariant polynomial functions on this vector space in terms of the category of representations. More precisely, we associate...

Hosting dynamic web services through proxies placed at the edge of the Internet is an upcoming trend that has the potential to scale these services to a very large number of geographically distributed clients. However, providing consistent access to shared mutable objects that make up the service, tolerating misbehaving proxies, and handling proxy disconnections, while still achieving the scala...

We prove a generating function formula for the Betti numbers of Nakajima quiver varieties. We prove that it is a q-deformation of the WeylKac character formula. In particular this implies that the constant term of the polynomial counting the number of absolutely indecomposable representations of a quiver equals the multiplicity of a certain weight in the corresponding Kac-Moody algebra, which w...

We discuss a general procedure to obtain 1/2 BPS partition functions for generic N = 1 quiver gauge theories. These functions count the gauge invariant operators (bosonic and fermionic), charged under all the global symmetries (mesonic and baryonic), in the chiral ring of a given quiver gauge theory. In particular we discuss the inclusion of the spinor degrees of freedom in the partition functi...

The aim of this paper is to develop an algebraic tool, namely the monoid structure on families of quiver representations as stated in the title, which is directly related both to quantized Kac-Moody algebras, and to geometric properties of varieties of representations. Although this new structure is far from being understood in detail, it it easily accessible to explicit calculations, and it pr...

We obtain a simple expression for the triangle ‘t Hooft anomalies in quiver gauge theories that are dual to toric Sasaki-Einstein manifolds. We utilize the result and simplify considerably the proof concerning the equivalence of a-maximization and Z-minimization. We also resolve the ambiguity in defining the flavor charges in quiver gauge theories. We then compare coefficients of the triangle a...

These are notes from introductory survey lectures given at the Institute for Studies in Theoretical Physics and Mathematics (IPM), Teheran, in 2008 and 2010. We present the definition and the fundamental properties of Fomin-Zelevinsky’s cluster algebras. Then we introduce quiver representations and show how they can be used to construct cluster variables, which are the canonical generators of c...

We show that many cluster-theoretic properties of the Markov quiver hold also for adjacency quivers of triangulations of once-punctured closed surfaces of arbitrary genus. Along the way we consider the class P of quivers introduced by Kontsevich and Soibelman, characterize the mutation-finite quivers that belong to that class and draw some conclusions regarding non-degenerate potentials on them...

We construct an action of N = 2 affine A n quiver gauge model having non-canonical kinetic terms and equipped with electric and magnetic FI terms. N = 2 supersymmetry is shown to be broken to N = 1 spontaneously and N = 1 multiplets realized on the vacua are given. We also mention the models with different gauge groups. It is argued that the affine A 1 quiver gauge model provides a dynamical re...

Abstract. We deduce using the Ringel-Hall algebra approach explicit formulas for the cardinalities of some Grassmannians over a finite field associated to the Kronecker quiver. We realize in this way a quantification of the formulas obtained by Caldero and Zelevinsky for the Euler characteristics of these Grassmannians. We also present a recursive algorithm for computing the cardinality of ever...