Quiver Varieties and Path Realizations arising from Adjoint Crystals of type $A_n^{(1)}$

Abstract

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

This is accepted for publication by Transactions of the American Mathematical Society