Quiver varieties and path realizations arising from adjoint crystals of type 𝐴_{𝑛}⁽¹⁾

Abstract

Let B ( Λ 0 ) B(\Lambda _0) be the level 1 highest weight crystal of the quantum affine algebra U q ( A n ( 1 ) ) U_q(A_n^{(1)}) . We construct an explicit crystal isomorphism between the geometric realization B ( Λ 0 ) \mathbb {B}(\Lambda _0) of B ( Λ 0 ) B(\Lambda _0) via quiver varieties and the path realization P ad ( Λ 0 ) {\mathcal P}^{\textrm {ad}}(\Lambda _0) of B ( Λ 0 ) B(\Lambda _0) arising from the adjoint crystal B ad B^{\textrm {ad}} .