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

TL;DR

This paper constructs a positive geometric crystal for the affine Lie algebra D_6^{(1)} at its spin node, advancing the understanding of geometric crystals in affine Lie algebra theory.

Contribution

It provides the first explicit construction of a positive geometric crystal for D_6^{(1)} at the spin node, confirming a conjecture for this case.

Findings

Constructed a positive geometric crystal for D_6^{(1)} at the spin node.

Supports the conjecture that each Dynkin node has an associated positive geometric crystal.

Enhances the understanding of geometric crystals in affine Lie algebra representations.

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 $k= 6$.