Compatibility degree of cluster complexes

TL;DR

This paper introduces a new compatibility degree function for cluster complexes based on $f$-vectors, generalizing the classical concept, and explores its properties and conjectures across various cluster algebra types.

Contribution

It defines a new compatibility degree for cluster complexes, proves several of its fundamental properties, and establishes the exchangeability property in specific cases.

Findings

Compatibility degree has duality, symmetry, embedding, and compatibility properties.

Proven exchangeability property for rank 2, acyclic skew-symmetric, and certain geometric cluster algebras.

Conjecture that compatibility degree has exchangeability property in general.

Abstract

We introduce a new function on the set of pairs of cluster variables via $f$-vectors, which we call it the compatibility degree (of cluster complexes). The compatibility degree is a natural generalization of the classical compatibility degree introduced by Fomin and Zelevinsky. In particular, we prove that the compatibility degree has the duality property, the symmetry property, the embedding property and the compatibility property, which the classical one has. We also conjecture that the compatibility degree has the exchangeability property. As pieces of evidence of this conjecture, we establish the exchangeability property for cluster algebras of rank 2, acyclic skew-symmetric cluster algebras, cluster algebras arising from weighted projective lines, and cluster algebras arising from marked surfaces.