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

TL;DR

This paper constructs an explicit isomorphism between geometric and path realizations of the level 1 highest weight crystal for quantum affine algebra of type A_n^{(1)}, linking quiver varieties and adjoint crystals.

Contribution

It provides a new explicit crystal isomorphism connecting geometric quiver variety realizations with path models from adjoint crystals for type A_n^{(1)}.

Findings

Established an explicit isomorphism between geometric and path realizations.

Unified geometric and combinatorial models of crystals for affine type A.

Enhanced understanding of crystal structures via quiver varieties and adjoint crystals.

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 ${\mathcal P}^{\rm ad}(\Lambda_0)$ of $B(\Lambda_0)$ arising from the adjoint crystal $\adjoint$.