Nakajima Varieties and Repetitive Algebras

Lecture 1: Nakajima Quiver Varieties

Recall that an algebraic group G is called (linearly) reductive if any its rational (i.e., algebraic) representation is completely reducible. The finite groups, the group GLn and the products GLn1 × . . .GLnk are reductive. Below G denotes a reductive algebraic group and X is an affine algebraic variety equipped with an (algebraic) action of G. Results explained below in this section can be fou...

Kac conjecture from Nakajima quiver varieties

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...

Kac’s conjecture from Nakajima quiver varieties

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...

Homological realization of Nakajima varieties and Weyl group actions

Nakajima Monomials and Crystals for Special Linear Lie Algebras

The theory of Nakajima monomials is a combinatorial scheme for realizing crystal bases of quantum groups. Nakajima introduced a certain set of monomials realizing the irreducible highest weight crystals in [16]. Kashiwara and Nakajima independently defined a crystal structure on the set of Nakajima monomials and also gave a realization of irreducible highest weight crystal B(λ) in terms of Naka...