2019/11/17 by Changjian Fu, Fu, Changjian, Yasuaki Gyoda +1
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Nonlinear Waves and Solitons #math.CO #math.RA #math.RT
paper · pdf · doi:10.48550/arxiv.1911.07193
37 pages, corrections to some definitions in Section 2.3, other minor corrections, accepted for publication in Annales de l'Institut Fourier
arxiv created 2021/12/20 · arxiv updated 2021/12/21
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.