Affine Geometric Crystal of type $D_4^{(3)}$

TL;DR

This paper explicitly constructs an affine geometric crystal of type D4^{(3)} linked to a fundamental representation, demonstrating its positive structure through an explicit realization.

Contribution

It provides the first explicit realization of an affine geometric crystal of type D4^{(3)} with a positive structure, advancing understanding of geometric crystals in this context.

Findings

Explicit realization of the affine geometric crystal of type D4^{(3)}

Demonstration of positive structure in the crystal

Connection to fundamental representation W(π_1)

Abstract

We shall realize certain affine geometric crystal of type $D_4^{(3)}$ associated with the fundamental representation $W(\pi_1)$ explicitly . By its explicit form, we see that it has a positive structure.