$D_6^{(1)}$- Geometric Crystal at the spin node

Abstract: Let $\mathfrak{g}$ be an affine Lie algebra with index set $I = {0, 1, 2, \cdots , n}$. It is conjectured that for each Dynkin node $k \in I \setminus {0}$ the affine Lie algebra $\mathfrak{g}$ has a positive geometric crystal. In this paper we construct a positive geometric crystal for the affine Lie algebra $D_6^{(1)}$ corresponding to the Dynkin spin node $k= 6$.