$D_5^{(1)}$- Geometric Crystal corresponding to the Dynkin spin node $i=5$ and its ultra-discretization

Abstract

Let $g$ be an affine Lie algebra with index set $I=\{0,1,2,\cdots ,n\}$ and $g^L$ be its Langlands dual. It is conjectured that for each Dynkin node $i\in I\setminus \{0\}$ the affine Lie algebra $g$ has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of certain coherent family of perfect crystals for $g^L$. In this paper we construct a positive geometric crystal $V(D_5^{(1)})$ in the level zero fundamental spin $D_5^{(1)}$- module $W({\varpi }_5)$. Then we define explicit $0$-action on the level $l$ known $D_5^{(1)}$- perfect crystal $B^{5,l}$ and show that $\{B^{5,l}{\}}_{l\ge 1}$ is a coherent family of perfect crystals with limit $B^{5,\infty }$. Finally we show that the ultra-discretization of $V(D_5^{(1)})$ is isomorphic to $B^{5,\infty }$ as crystals which proves the conjecture in this case.

25 pages