On Cₙ⁽¹⁾-Geometric Crystal and its Ultradiscretization

Let g be an affine Lie algebra with index set I = \0, 1, 2, ⋯ , n\ and gL be its Langlands dual. It is conjectured that for each Dynkin node i ∈ I \0\ the affine Lie algebra g has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of a certain coherent family of perfect crystals for the Langland dual gL. In this paper we construct positive geometric crystals for V(Cₙ⁽¹⁾) in the level zero fundamental spin Cₙ⁽¹⁾- module W(ₙ) for $n = 2, 3,4$ and show that its ultra-discretization is isomorphic to the limit Bn, ∞ of a coherent family , l ≥ 1 of perfect crystals for the Langland dual Dₙ⁽²⁾ which proves the conjecture in these cases.