Lattice structure in cluster algebra of finite type and non-simply-laced Ingalls-Thomas bijection

TL;DR

This paper reveals that the lattice structure of clusters in finite type cluster algebras is anti-isomorphic to certain torsion and Cambrian lattices, linking combinatorial and algebraic structures.

Contribution

It establishes explicit anti-isomorphisms between cluster lattice structures and torsion and Cambrian lattices via exchange quivers and support τ-tilting modules.

Findings

Cluster lattice is anti-isomorphic to torsion lattice of GLS path algebra.

Exchange quivers are anti-isomorphic to support τ-tilting modules.

Cluster structures correspond to c-clusters of almost positive roots.

Abstract

In this paper, we demonstrate that the lattice structure of a set of clusters in a cluster algebra of finite type is anti-isomorphic to the torsion lattice of a certain Geiss-Leclerc-Schr\"oer (GLS) path algebra and to the $c$-Cambrian lattice. We prove this by explicitly describing the exchange quivers of cluster algebras of finite type. Specifically, we prove that these quivers are anti-isomorphic to those formed by support $\tau$-tilting modules in GLS path algebras and to those formed by $c$-clusters consisting of almost positive roots.