Ultra-discretization of the D_4^3-Geometric Crystals to the G_2^1-Perfect Crystals

TL;DR

This paper proves that the ultra-discretization of a specific positive geometric crystal for the affine Lie algebra D_4^3 is isomorphic to the limit of a coherent family of perfect crystals for its Langlands dual G_2^1, confirming a conjecture.

Contribution

It establishes the isomorphism between the ultra-discretized geometric crystal for D_4^3 and the limit of perfect crystals for G_2^1, providing evidence for the conjectured relationship.

Findings

Ultra-discretization of D_4^3 geometric crystal matches the G_2^1 perfect crystal limit.

Confirms the conjecture relating geometric crystals and perfect crystals via ultra-discretization.

Provides a concrete example supporting the geometric crystal and Langlands duality connection.

Abstract

Let g be an affine Lie algebra and g^L be its Langlands dual. It is conjectured that 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 that the ultra-discretization of the positive geometric crystal for g = D_4^3 given by Igarashi and Nakashima is isomorphic to the limit of the coherent family of perfect crystals for g^L= G_2^1 constructed recently by Misra, Mohamad and Okado.