A short proof for the acyclicity of oriented exchange graphs of cluster algebras
Shuhao Deng, Changjian Fu
TL;DR
This work provides a concise proof of the acyclicity of the oriented exchange graph for cluster algebras by leveraging the cluster scattering diagram framework. It situates the problem within the established machinery of seeds, $C$- and $G$-matrices, sign-coherence, and green mutations, and then applies the GHKK cluster scattering diagram $\mathfrak{D}_0(B)$ along with tropical duality to rule out directed cycles. The key argument asserts that any potential cycle would yield a nontrivial element in the structure group that cannot vanish under finite truncation, leading to a contradiction. The result streamlines the proof of acyclicity and highlights the utility of scattering diagrams in the combinatorial topology of cluster algebras.
Abstract
The statement in the title was proved in \cite{Cao23} by introducing dominant sets of seeds, which are analogs of torsion classes in representation theory. In this note, we observe a short proof by the existence of consistent cluster scattering diagrams.