$A_n^{(1)}$-Geometric Crystal corresponding to Dynkin index $i=2$ and its ultra-discretization

TL;DR

This paper proves a conjecture linking positive geometric crystals and perfect crystals for the affine Lie algebra $A_n^{(1)}$, specifically for the case when the Dynkin index is 2, through ultra-discretization.

Contribution

It establishes the conjecture for $A_n^{(1)}$ with index $i=2$, demonstrating the isomorphism between ultra-discretized geometric crystals and limits of perfect crystals.

Findings

Confirmed the conjecture for $A_n^{(1)}$ at $i=2$

Established the isomorphism between ultra-discretized crystals and perfect crystal limits

Extended understanding of geometric and combinatorial crystal relations

Abstract

Let $g$ be an affine Lie algebra with index set $I = \{0, 1, 2,..., n\}$ and $g^L$ be its Langlands dual. It is conjectured that for each $i \in I \setminus \{0\}$ the affine Lie algebra $g$ has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of certain coherent family of perfect crystals for $g^L$. We prove this conjecture for $i=2$ and $g = A_n^{(1)}$.