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 $\{ B^{ 5, l}\}_{ l \geq 1}$ 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.
Categories: math.QA