Abstract
Lots of research focuses on the combinatorics behind various bases of cluster algebras. This paper studies the natural basis of a type $A$ cluster algebra, which consists of all cluster monomials. We introduce a new kind of combinatorial formula for the cluster monomials in terms of the so-called globally compatible collections. We give bijective proofs of these formulas by comparing with the well-known combinatorial models of the $T$-paths and of the perfect matchings in a snake diagram. For cluster variables of a type $A$ cluster algebra, we give a bijection that relates our new formula with the theta functions constructed by Gross, Hacking, Keel and Kontsevich.
Highlights
Cluster algebras were first introduced by S
A cluster algebra is a subring of a rational function field generated by a distinguished set of Laurent polynomials called cluster variables
For cluster variables of a type A quiver, we construct a bijection between GCSs and broken lines in Theorem 7.10 and 7.13, which relates our new formula with the theta functions constructed in [10]
Summary
Cluster algebras were first introduced by S. For cluster variables of a type A quiver, we construct a bijection between GCSs (which is equivalent to GCCs) and broken lines in Theorem 7.10 (the even rank case) and 7.13 (the general case), which relates our new formula with the theta functions constructed in [10]. The simplicity of this bijection came as a surprise for us: namely, under our setting, the i-th number in a GCS (which is a 0-1 sequence) is 0 if and only if the corresponding broken line bends at the i-th coordinate hyperplane e⊥i. We give another proof of Theorem 4.6 using T -paths
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