Quiver varieties and adjoint crystals of level $\ell$ for type $A_n^{(1)}$

TL;DR

This paper establishes an explicit isomorphism between geometric and combinatorial realizations of level  highest weight crystals for quantum affine algebra of type A^{(1)}, linking quiver varieties with adjoint crystal paths.

Contribution

It constructs a detailed crystal isomorphism connecting geometric quiver variety models with path models from the adjoint crystal for type A^{(1)}.

Findings

Explicit crystal isomorphism constructed

Bridges geometric and combinatorial models

Enhances understanding of crystal structures in affine types

Abstract

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