$D_6^{(1)}$- 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 $k=6$.

19 pages. arXiv admin note: substantial text overlap with arXiv:1812.01651