Ultra-Discretization of $D_6^{(1)}$- 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 in \cite{KNO} that for each Dynkin node $k\in I\setminus \{0\}$ the affine Lie algebra $\mathfrak{g}$ has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of a coherent family of perfect crystals for the Langland dual ${\mathfrak{g}}^L$. In this paper we show that at the spin node $k=6$, the family of perfect crystals given in \cite{KMN2} form a coherent family and show that its limit $B^{6,\infty }$ is isomorphic to the ultra-discretization of the positive geometric crystal we constructed in \cite{MP} for the affine Lie algebra $D_6^{(1)}$ which proves the conjecture in this case.

arXiv admin note: substantial text overlap with arXiv:1812.01651

References in corpus (1)

• Existence of Kirillov-Reshetikhin crystals for nonexceptional types