$D_6^{(1)}$ -几何晶体自旋节点的超离散化
Ultra-Discretization of $D_6^{(1)}$- Geometric Crystal at the spin node
摘要：
设$\mathfrak g$是具有指标集$I = \{0, 1, 2, \cdots , n\}$的仿射李代数.在\cite{KNO}中证明了对于每个Dynkin节点$k \in I \setminus \{0\}$，仿射李代数$\mathfrak g$都有一个正几何晶体，其超离散化同构于Langland对偶${\mathfrak g} ^L$的一个相干完美晶体族的极限.本文证明了在自旋结点$k=6$处，\cite{KMN2}中给出的理想晶体族形成一个相干族，并证明了它的极限$B^{6,\infty}$同构于我们在\cite{MP}中构造的仿射李代数$D_6^{(1)}$的正几何晶体的超离散化，从而证明了在这种情况下的猜想.
英文摘要：
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.
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