Ultra-discretization of the D₄⁽³⁾-geometric crystal to the G₂⁽¹⁾-perfect crystals

Mana Igarashi, Kailash C. Misra and Toshiki Nakashima

Vol. 255 (2012), No. 1, 117–142

DOI: 10.2140/pjm.2012.255.117

Abstract

Let g be an affine Lie algebra and g^L its Langlands dual. It was conjectured by Kashiwara, Nakashima, and Okado 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 = D₄⁽³⁾ given by Igarashi and Nakashima is isomorphic to the limit of the coherent family of perfect crystals for g^L = G₂⁽¹⁾ constructed by Misra, Mohamad, and Okado.

Keywords

geometric crystals, perfect crystals, ultra-discretization

Mathematical Subject Classification 2010

Primary: 17B37, 17B67

Secondary: 22E65, 14M15

Milestones

Received: 5 October 2010

Accepted: 3 May 2011

Published: 14 March 2012