Affine Geometric Crystal of $A^{(1)}_n$ and Limit of Kirillov-Reshetikhin Perfect Crystals

TL;DR

This paper constructs a positive geometric crystal for the affine Lie algebra $A^{(1)}_n$, confirming a conjecture by demonstrating its ultra-discretization matches the limit of certain perfect crystals, using lattice-path combinatorics.

Contribution

It provides the first explicit construction of a positive geometric crystal for $A^{(1)}_n$ and verifies its ultra-discretization corresponds to the limit of perfect crystals, supporting the conjecture.

Findings

Constructed positive geometric crystals for $A^{(1)}_n$.

Proved ultra-discretization matches the limit of perfect crystals.

Utilized lattice-path combinatorics in the construction.

Abstract

Let $\mathfrak g$ be an affine Lie algebra with index set $I = \{0, 1, 2, \cdots , n\}$ and ${\mathfrak g}^L$ be its Langlands dual. It is conjectured by Kashiwara et al.([16]) that for each $k \in I \setminus \{0\}$ the affine Lie algebra $\mathfrak g$ has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of certain coherent family of perfect crystals for ${\mathfrak g}^L$. Motivated by this conjecture we construct a positive geometric crystal for the affine Lie algebra ${\mathfrak g}= A^{(1)}_n$ for each Dynkin index $k\in I\setminus\{0\}$ and show that its ultra-discretization is isomorphic to the limit of a coherent family of perfect crystals for $A^{(1)}_n$ given by Okado et al.([29]). In the process we develop and use some lattice-path combinatorics.