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

摘要

Let g be an affine Lie algebra and g^L be its Langlands dual. It isconjectured that g has a positive geometric crystal whose ultra-discretizationis isomorphic to the limit of certain coherent family of perfect crystals forg^L. We prove that the ultra-discretization of the positive geometric crystalfor g = D_4^3 given by Igarashi and Nakashima is isomorphic to the limit of thecoherent family of perfect crystals for g^L= G_2^1 constructed recently byMisra, Mohamad and Okado.