Nakajima quiver varieties, affine crystals and combinatorics of Auslander-Reiten quivers

TL;DR

This paper establishes an explicit isomorphism between geometric and combinatorial realizations of crystal bases for certain Lie algebra representations, linking Nakajima quiver varieties with Auslander-Reiten quivers.

Contribution

It provides a new explicit crystal isomorphism connecting geometric and combinatorial models for type A and D Lie algebra representations.

Findings

Explicit crystal isomorphism between geometric and combinatorial models

Homological description of irreducible components of Lusztig's quiver varieties

Geometric realization of Kirillov-Reshetikhin crystals

Abstract

We obtain an explicit crystal isomorphism between two realizations of crystal bases of finite dimensional irreducible representations of simple Lie algebras of type A and D. The first realization we consider is a geometric construction in terms of irreducible components of certain Nakajima quiver varieties established by Saito and the second is a realization in terms of isomorphism classes of quiver representations obtained by Reineke. We give a homological description of the irreducible components of Lusztig's quiver varieties which correspond to the crystal of a finite dimensional representation and describe the promotion operator in type A to obtain a geometric realization of Kirillov-Reshetikhin crystals.