Minimal Matchings for dP3 Cluster Variables

TL;DR

This paper extends combinatorial formulas for dP3 cluster variables by introducing a deformed quiver with a second variable set, utilizing Aztec castles and minimal matchings to generalize generating functions.

Contribution

It introduces a novel deformation of the dP3 quiver with principal coefficients, providing explicit formulas for cluster variables via minimal matchings of Aztec castles.

Findings

Explicit formulas for deformed dP3 cluster variables

Extension of known generating functions using Aztec castles

Connection between minimal matchings and cluster algebra deformations

Abstract

In previous work [LM17], Tri Lai and the second author studied a family of subgraphs of the dP3 brane tiling, called Aztec castles, whose dimer partition functions provide combinatorial formulas for cluster variables resulting from mutations of the quiver associated with the del Pezzo surface dP3. In our paper, we investigate a variant of the dP3 quiver by considering a second alphabet of variables that breaks the symmetries of the relevant recurrences. This deformation is motivated by the theory of cluster algebras with principal coefficients introduced by Fomin and Zelevinsky. Our main result gives an explicit formula extending previously known generating functions for dP3 cluster variables by using Aztec castles and constructing their associated minimal matchings.