Gale-Robinson Quivers and Principal Coefficients

TL;DR

This paper provides a combinatorial interpretation of Laurent polynomials from periodic quiver mutations with frozen vertices, linking Gale-Robinson sequences to cluster variables with principal coefficients.

Contribution

It introduces a new combinatorial framework for understanding Gale-Robinson sequences within cluster algebra theory, completing previous partial results.

Findings

Laurent polynomials are interpreted combinatorially via quiver mutations.

Gale-Robinson sequences are connected to cluster variables with principal coefficients.

The work extends and completes earlier partial results in the field.

Abstract

In this paper, we provide a combinatorial interpretation for Laurent polynomials obtained by iteratively mutating a certain periodic quiver that has been framed with frozen vertices. This yields a family of cluster variables with principal coefficients associated to a family of integer sequences known as Gale-Robinson sequences. The work of this paper completes arguments for preliminary results announced in earlier work of Jeong-Musiker-Zhang, and relates to works of Bousquet-M\'elou-Propp-West, Speyer, Vichitkunakorn, and of Eager-Franco.