A Combinatorial Formula for Certain Elements of Upper Cluster Algebras

TL;DR

This paper introduces a simple formula for specific positive elements in upper cluster algebras, demonstrating basis formation in acyclic cases and inclusion properties in non-acyclic rank 3 cases.

Contribution

It provides an elementary combinatorial formula for certain elements and establishes basis and containment results in cluster algebra theory.

Findings

Elements with positive coefficients form a basis in acyclic cluster algebras.

Non-acyclic rank 3 cluster algebras are properly contained in their upper cluster algebras.

The formula simplifies understanding of element structure in upper cluster algebras.

Abstract

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.