A Combinatorial Formula for Certain Elements of Upper Cluster Algebras

[摘要] We develop an elementary formula for certain non-trivial elements of upper cluster algebras. These elements have positive coefficients. We show that when the cluster algebra is acyclic these elements form a basis. Using this formula, we show that each non-acyclic skew-symmetric cluster algebra of rank 3 is properly contained in its upper cluster algebra.

[关键词] cluster algebra;upper cluster algebra;Dyck path