Linear independence of cluster monomials for skew-symmetric cluster algebras

TL;DR

This paper proves that in skew-symmetric cluster algebras, cluster monomials are linearly independent and the exchange graph and cluster complex do not depend on coefficients, confirming longstanding conjectures.

Contribution

The paper establishes the linear independence of cluster monomials and coefficient independence of exchange structures specifically for skew-symmetric cluster algebras.

Findings

Confirmed linear independence of cluster monomials in skew-symmetric cluster algebras

Proved exchange graph and cluster complex are coefficient-independent

Validated conjectures proposed by Fomin-Zelevinsky

Abstract

Fomin-Zelevinsky conjectured that in any cluster algebra, the cluster monomials are linearly independent and that the exchange graph and cluster complex are independent of the choice of coefficients. We confirm these conjectures for all skew-symmetric cluster algebras.