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

Abstract

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 $\{B^{n,l}{\}}_{l\ge 1}$ of perfect crystals for the Langland dual $D_n^{(2)}$ which proves the conjecture in these cases.