D5(1)-Geometric crystal corresponding to the Dynkin spin node i = 5 and its ultra-discretization

Abstract

Let [Formula: see text] be an affine Lie algebra with index set [Formula: see text] and [Formula: see text] be its Langlands dual. It is conjectured that for each Dynkin node [Formula: see text] the affine Lie algebra [Formula: see text] has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of certain coherent family of perfect crystals for [Formula: see text]. In this paper, we construct a positive geometric crystal [Formula: see text] in the level zero fundamental spin [Formula: see text]-module [Formula: see text]. Then we define explicit [Formula: see text]-action on the level [Formula: see text] known [Formula: see text]-perfect crystal [Formula: see text] and show that [Formula: see text] is a coherent family of perfect crystals with limit [Formula: see text]. Finally, we show that the ultra-discretization of [Formula: see text] is isomorphic to [Formula: see text] as crystals which proves the conjecture in this case.