Stasheff polytopes and the coordinate ring of the cluster X-variety of type A_n

TL;DR

This paper introduces Stasheff polytopes within the tropical space of cluster A-varieties and demonstrates their role as supports of products of canonical basis elements in type A_n cluster X-varieties.

Contribution

It establishes a novel connection between Stasheff polytopes and the supports of canonical basis products in cluster X-varieties of type A_n.

Findings

Supports of products of canonical basis elements are Stasheff polytopes.

Defines Stasheff polytopes in the tropical space of cluster A-varieties.

Proves the correspondence specifically for type A_n cluster X-varieties.

Abstract

We define Stasheff polytopes in the space of tropical points of cluster A-varieties. We study the supports of products of elements of canonical bases for cluster X-varieties. We prove that, for the cluster X-variety of type A_n, such supports are Stasheff polytopes.