Quivers are directed graphs. The term is the term used in representation theory, which goes along with the following notion: a representation of a quiver is an assignment of vector spaces to vertices and linear maps between the vector spaces to the arrows. Quivers appear in many areas of mathematics: (1) Algebraic geometry (Hilbert schemes, moduli spaces (represent these as varieties of quiver ...

We study algebraic actions of finite groups of quiver automorphisms on moduli spaces of quiver representations. We decompose the fixed loci using group cohomology and we give a modular interpretation of each component. As an application, we construct branes in hyperkähler quiver varieties, as fixed loci of such actions.

We define a functor which gives the “global rank of a quiver representation” and prove that it has nice properties which make it a generalization of the rank of a linear map. We demonstrate how to construct other “rank functors” for a quiver Q, which induce ring homomorphisms (called “rank functions”) from the representation ring of Q to Z. These rank functions give discrete numerical invariant...

We construct Nakajima’s quiver varieties of type A in terms of affine Grassmannians of type A. This gives a compactification of quiver varieties and a decomposition of affine Grassmannians into a disjoint union of quiver varieties. Consequently, singularities of quiver varieties, nilpotent orbits and affine Grassmannians are the same in type A. The construction also provides a geometric framewo...

Using subvarieties, which we call Demazure quiver varieties, of the quiver varieties of Nakajima, we give a geometric realization of Demazure modules of Kac-Moody algebras with symmetric Cartan data. We give a natural geometric characterization of the extremal weights of a representation and show that Lusztig's semicanonical basis is compatible with the filtration of a representation by Demazur...

Inspired by Cachazo, Katz and Vafa (“Geometric transitions and N = 1 quiver theories” (hep-th/0108120)), we examine representations of “N = 1 quivers” arising from string theory. We derive some mathematical consequences of the physics, and show that these results are a natural extension of Gabriel’s ADE theorem. Extending the usual ADE case that relates quiver representations to curves on surfa...

Given a basic K-coalgebra C, we study the left Gabriel-valued quiver (CQ,Cd) of C by means of irreduciblemorphisms between indecomposable injective leftC-comodules and by means of the powers rad of the radical rad of the category C-inj of the socle-finite injective left C-comodules. Connections between the valued quiver (CQ,C d) of C and the valued quiver (CQ,Cd) of a colocalization coalgebra q...

In this paper we detail the lattice constructions of several classes of supersymmetric quiver gauge theories in two and three Euclidean spacetime dimensions possessing exact supersymmetry at finite lattice spacing. Such constructions are obtained through the methods of topological twisting and geometric discretization of Euclidean Yang-Mills theories with eight and sixteen supercharges in two a...

In a previous paper the authors have attached to each Dynkin quiver an associative algebra. The definition is categorical and the algebra is used to construct desingularizations of arbitrary quiver Grassmannians. In the present paper we prove that this algebra is isomorphic to an algebra constructed by Hernandez-Leclerc defined combinatorially and used to describe certain graded Nakajima quiver...

We give four positive formulae for the (equioriented type A) quiver polynomials of Buch and Fulton [BF99]. All four formulae are combinatorial, in the sense that they are expressed in terms of combinatorial objects of certain types: Zelevinsky permutations, lacing diagrams, Young tableaux, and pipe dreams (also known as rc-graphs). Three of our formulae are multiplicity-free and geometric, mean...