The Discrete Fundamental Group of the Associahedron, and the Exchange Module

TL;DR

This paper explores the discrete fundamental group of the associahedron and introduces the exchange module for type A_n cluster algebras, revealing their algebraic structures and bases.

Contribution

It provides a new discrete homotopy perspective on the associahedron and defines the exchange module, linking topological and algebraic properties in cluster algebra theory.

Findings

Discrete fundamental group is free abelian of rank C(n+2,4).

Exchange module is free abelian of rank C(n+2,3).

Combinatorial basis descriptions are provided.

Abstract

The associahedron is an object that has been well studied and has numerous applications, particularly in the theory of operads, the study of non-crossing partitions, lattice theory and more recently in the study of cluster algebras. We approach the associahedron from the point of view of discrete homotopy theory. We study the abelianization of the discrete fundamental group, and show that it is free abelian of rank $\binom{n+2}{4}$. We also find a combinatorial description for a basis of this rank. We also introduce the exchange module of the type $A_n$ cluster algebra, used to model the relations in the cluster algebra. We use the discrete fundamental group to the study of exchange module, and show that it is also free abelian of rank $\binom{n+2}{3}$.