Compatibility degree of cluster complexes
Compatibility degree of cluster complexes
复制标题
DOI：
10.5802/aif.3596
复制
发表时间：
2019-11
期刊：
影响因子：
--
通讯作者：
Changjian Fu;Y. Gyoda
中科院分区：
文献类型：
--
作者：
Changjian Fu;Y. Gyoda
我们通过$f$-向量在簇变量对的集合上引入了一个新的函数，我们称之为（簇复合物的）相容度。相容度是Fomin和Zelevinsky提出的经典相容度的自然推广。特别地，我们证明了相容度具有经典相容度所具有的对偶性、对称性、嵌入性和相容性。我们还推测相容度具有交换性。作为这一猜想的证据，我们建立了秩为2的簇代数、无环斜对称簇代数、由加权射影线产生的簇代数和由标记曲面产生的簇代数的交换性质，除了一次穿孔闭曲面.
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 except for the once-punctured closed surfaces.