Ordered Exchange Graphs

TL;DR

This paper introduces the concept of ordered exchange graphs in cluster algebras, generalizing and axiomatizing the combinatorial structures arising from representation theory and tilting theory.

Contribution

It formalizes the structure of ordered exchange graphs, unifies various constructions from representation theory, and relates them to the poset of tilting modules.

Findings

Ordered exchange graphs generalize tilting module posets.

Various constructions of exchange graphs are related through axiomatization.

The framework unifies combinatorial and representation-theoretic perspectives.

Abstract

The exchange graph of a cluster algebra encodes the combinatorics of mutations of clusters. Through the recent "categorifications" of cluster algebras using representation theory one obtains a whole variety of exchange graphs associated with objects such as a finite-dimensional algebra or a differential graded algebra concentrated in non-positive degrees. These constructions often come from variations of the concept of tilting, the vertices of the exchange graph being torsion pairs, t-structures, silting objects, support $\tau$-tilting modules and so on. All these exchange graphs stemming from representation theory have the additional feature that they are the Hasse quiver of a partial order which is naturally defined for the objects. In this sense, the exchange graphs studied in this article can be considered as a generalization or as a completion of the poset of tilting modules which has been studied by Happel and Unger. The goal of this article is to axiomatize the thus obtained structure of an ordered exchange graph, to present the various constructions of ordered exchange graphs and to relate them among each other.