The ubiquitous A-D-E classification has induced many proposals of often mysterious correspondences both in mathematics and physics. The mathematics side includes quiver theory and the McKay Correspondence which relates finite group representation theory to Lie algebras as well as crepant resolutions of Gorenstein singularities. On the physics side, we have the graph-theoretic classification of ...

For irreducible integrable highest weight modules of the finite and affine Lie algebras of type A and D, we define an isomorphism between the geometric realization of the crystal graphs in terms of irreducible components of Nakajima quiver varieties and the combinatorial realizations in terms of Young tableaux and Young walls. For type A (1) n , we extend the Young wall construction to arbitrar...

In 1979, Norton showed that the representation theory of the 0-Hecke algebra admits a rich combinatorial description. Her constructions rely heavily on some triangularity property of the product, but do not use explicitly that the 0-Hecke algebra is a monoid algebra. The thesis of this paper is that considering the general setting of monoids admitting such a triangularity, namely J -trivial mon...

A bstract We derive a generalized Knizhnik-Zamolodchikov equation for the correlation function of intertwiners vector and MacMahon representations Ding-Iohara-Miki algebra. These are cousins refined topological vertex which is regarded as intertwining operator Fock representation. The shift spectral parameter generated by constructed from universal R matrix. solutions to KZ factorized into rati...

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...