A NEW COMBINATORIAL FORMULA FOR CLUSTER MONOMIALS OF EQUIORIENTED TYPE A QUIVERS

Abstract. Cluster algebras were first introduced by S. Fomin and A. Zelevinsky in 2001. Since then, an important question has been how to explicitly construct good bases in them, that is, bases having positive structure constants, containing the cluster monomials, etc. There currently exists a formula for the cluster variables of cluster algebras associated to polygons in terms of T-paths, which has been modified and generalized to cluster algebras coming from surfaces. However, this formula does not seem to directly generalize to arbitrary cluster algebras. In this paper, we give a new (and more compact) combinatorial formula for all cluster monomials of the cluster algebra associated to any equioriented type A quiver, in terms of compatible subpaths on maximal Dyck paths. This formula has the potential to generalize to cluster algebras beyond the ones coming strictly from surfaces. 1. introduction A (skew-symmetric) cluster algebra A is a subalgebra of a rational function field with a distinguished set of variables (cluster variables), grouped into overlapping subsets (clusters), and defined by a recursive procedure (mutation) on quivers. A cluster monomial is a mono-mial in the elements of any cluster. It is expected that each cluster algebra admits a “good” basis, that is, a basis having positive structure constants, containing the cluster monomials, etc. Caldero and Keller [1] showed that the cluster monomials form a basis of the cluster algebra if and only if there is only a finite number of cluster variables. Recently Cerulli Irelli, Keller, Labardini-Fragoso and Plamondon [2] proved that the cluster monomials are linearly independent for all cluster algebras associated to quivers. Given any quiver Q without loops and 2-cycles, a unique (coefficient-free) cluster algebra A(Q) (up to isomorphism) can be defined. In this paper we deal with the quive