This paper proves that every projective toric variety is the fine moduli space for stable representations of an appropriate bound quiver. To accomplish this, we study the quiver Q with relations R corresponding to the finite-dimensional algebra End ( ⊕r i=0 Li) where L := (OX , L1, . . . , Lr) is a list of line bundles on a projective toric variety X. The quiver Q defines a smooth projective to...

Abstract Quiver representations arise naturally in many areas across mathematics. Here we describe an algorithm for calculating the vector space of sections, or compatible assignments vectors to vertices, any finite-dimensional representation a finite quiver. Consequently, are able define and compute principal components with respect quiver representations. These solutions constrained optimisat...

We prove a Chevalley restriction theorem and its double analogue for the cyclic quiver. The aim of this paper is to prove a Chevalley restriction theorem and its double analogue for the cyclic quiver. When the quiver is of type Â0, we recover the results for gln. The proof of our Chevalley restriction theorem is similar to the proof for gln; however, the proof of the double analogue uses a theo...

We present Quiver, a system that coordinates service proxies placed at the “edge” of the Internet to serve distributed clients accessing a service involving mutable objects. Quiver enables these proxies to perform consistent accesses to shared objects, by migrating the objects to proxies performing operations on those objects. These migrations dramatically improve performance when operations in...

One can geometrically engineer supersymmetric field theories theories by placing D-branes at or near singularities. The opposite process is described, where one can reconstruct the singularities from quiver theories. The description is in terms of a noncommutative quiver algebra which is constructed from the quiver diagram and the superpotential. The center of this noncommutative algebra is a c...

We consider representations of quivers in arbitrary categories and twisted representations of quivers in arbitrary tensor categories. We show that if A is an abelian category, then the category of representations of a quiver in A is also abelian, and that the category of twisted linear representations of a quiver is equivalent to the category of linear (untwisted) representations of a different...

We saw in lectures 7 and 8 how Lusztig’s nilpotent variety can be used to realize U−(g) and the crystal B(∞). Last week we saw how to use quiver grassmannians to realize the highest weight modules V (λ) as a quotient of U−(g), and the same construction realizes the crystals B(λ). This week we discuss a more standard approach to realizing V (λ) and B(λ), namely we will use Nakajima’s quiver vari...

We prove a generating function formula for the Betti numbers of Nakajima quiver varieties. We prove that it is a q-deformation of the Weyl-Kac 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 a certain weight in the corresponding Kac-Moody algebra, whic...

We show that the Higgs branch of a four-dimensional Yang-Mills theory, with gauge and matter content summarised by an ADE quiver diagram, is identical to the generalised Coulomb branch of a four-dimensional superconformal strongly coupled gauge theory with ADE global symmetry. This equivalence suggests the existence of a mirror symmetry between the quiver theories and the strongly coupled theor...