TL;DR
This paper develops new cluster expansion formulas for type A cluster algebras with boundary arc coefficients, using combinatorial and quiver-based methods, extending previous formulas by Musiker and Schiffler.
Contribution
It introduces three novel cluster expansion formulas for type A cluster algebras with boundary coefficients, utilizing perfect matchings, arrow subsets, and minimal cuts.
Findings
Three new cluster expansion formulas are established.
Formulas are based on perfect matchings, arrow subsets, and minimal cuts.
Results extend existing formulas to boundary arc coefficients.
Abstract
The aim of this paper is to give analogs of the cluster expansion formula of Musiker and Schiffler for cluster algebras of type A with coefficients arising from boundary arcs of the corresponding triangulated polygon. Indeed, we give three cluster expansion formulas by perfect matchings of angles in triangulated polygon, by discrete subsets of arrows of the corresponding ice quiver and by minimal cuts of the corresponding quiver with potential.
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