# $D_5^{(1)}$- Geometric Crystal corresponding to the Dynkin spin node $i=5$ and its ultra-discretization

## TL;DR

This paper constructs a positive geometric crystal for the affine Lie algebra $D_5^{(1)}$, explicitly describes its ultra-discretization, and proves it matches the limit of a coherent family of perfect crystals, confirming a conjecture in this case.

## Contribution

It provides the first explicit construction of a positive geometric crystal for $D_5^{(1)}$ and establishes its ultra-discretization as the limit of perfect crystals, confirming the conjecture for this algebra.

## Key findings

- Constructed a positive geometric crystal $V(D_5^{(1)})$ for $D_5^{(1)}$.
- Defined explicit $0$-action on the perfect crystal $B^{5, l}$.
- Proved the ultra-discretization of $V(D_5^{(1)})$ is isomorphic to $B^{5, \, ext{infinity}}$.

## Abstract

Let $g$ be an affine Lie algebra with index set $I = \{0, 1, 2, \cdots , n\}$ and $g^L$ be its Langlands dual. It is conjectured that for each Dynkin node $i \in I \setminus \{0\}$ the affine Lie algebra $g$ has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of certain coherent family of perfect crystals for $g^L$. In this paper we construct a positive geometric crystal $V(D_5^{(1)})$ in the level zero fundamental spin $D_5^{(1)}$- module $W(\varpi_5)$. Then we define explicit $0$-action on the level $l$ known $D_5^{(1)}$- perfect crystal $B^{5, l}$ and show that $\{B^{5, l}\}_{l \geq 1}$ is a coherent family of perfect crystals with limit $B^{5, \infty}$. Finally we show that the ultra-discretization of $V(D_5^{(1)})$ is isomorphic to $B^{5, \infty}$ as crystals which proves the conjecture in this case.