Epsilon Systems on Geometric Crystals of Type $A_n$

TL;DR

This paper introduces an epsilon system on geometric crystals of type A_n, demonstrating its product structure and invariance under tropical R maps, advancing understanding of algebraic and combinatorial properties.

Contribution

It presents a novel epsilon system on geometric crystals of type A_n, highlighting its algebraic structure and invariance properties not previously established.

Findings

Epsilon system defined on geometric crystals of type A_n

Product structure of the epsilon system established

Invariance under tropical R maps demonstrated

Abstract

We introduce an epsilon system on a geometric crystal of type $A_n$, which is a certain set of rational functions with some nice properties. We shall show that it is equipped with a product structure and that it is invariant under the action of tropical R maps.