Ultra-discretization of the G^(1)_2-Geometric Crystals to the D^(3)_4-Perfect Crystals

TL;DR

This paper proves that the ultra-discretization of a specific affine geometric crystal of type G^(1)_2 is isomorphic to the limit of a coherent family of perfect crystals of type D^(3)_4, confirming a prior conjecture.

Contribution

It establishes the isomorphism between the ultra-discretized G^(1)_2 geometric crystal and D^(3)_4 perfect crystals, confirming a conjecture in the field.

Findings

Confirmed the conjecture in [15] about the isomorphism.

Established the ultra-discretization as a bridge between geometric and perfect crystals.

Provided a new link between affine geometric crystals and perfect crystals of different types.

Abstract

We obtain the affirmative answer to the conjecture in [15]. More precisely, let X be the affine geometric crystal of type G^(1)_2 in [15] and UD(X,T,\theta) a ultra-discretization of X with respect to a certain positive structure \theta. Then we show that UD(X,T,\theta) is isomorphic to the limit of coherent family of perfect crystals of type D^(3)_4 in [7].