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

We obtain an explicit crystal isomorphism be-tween two realizations of crystal bases of finite dimensional irre-ducible 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 vari-eties 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