On $C_n^{(1)}$-Geometric Crystal and its Ultradiscretization

Let $\mathfrak{g}$ be an affine Lie algebra with index set $I = \{0, 1, 2, \cdots , n\}$ and $\mathfrak{g}^L$ be its Langlands dual. It is conjectured that for each Dynkin node $i \in I \setminus \{0\}$ the affine Lie algebra $\mathfrak{g}$ 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 $\mathfrak{g}^L$. In this paper we construct positive geometric crystals for $\mathcal{V}(C_n^{(1)})$ in the level zero fundamental spin $C_n^{(1)}$- module $W(\varpi_n)$ for $n = 2, 3,4$ and show that its ultra-discretization is isomorphic to the limit $B^{n, \infty}$ of a coherent family $\{B^{n, l}\}_{l \geq 1}$ of perfect crystals for the Langland dual $D_n^{(2)}$ which proves the conjecture in these cases.