Gale-Robinson Sequences and Brane Tilings

In-Jee Jeong,Sicong Zhang,Gregg Musiker

doi:10.46298/dmtcs.2336

Abstract

We study variants of Gale-Robinson sequences, as motivated by cluster algebras with principal coefficients. For such cases, we give combinatorial interpretations of cluster variables using brane tilings, as from the physics literature.

Highlights

This article is concerned with a variant of the Gale-Robinson integer sequence [Gal91], i.e. {xn} satisfying xnxn−N = xn−rxn−N+r + xn−sxn−N+s, where we include a second alphabet of variables, {y1, y2, . . . , yn}, that breaks the symmetry of this recurrence. This deformation is motivated by the theory of cluster algebras with principal coefficients
The undeformed version of this sequence has been studied by several authors [BPW09, FZ02b, S07]
We summarize how to go from a Gale-Robinson quiver, QN(r,s), to an associated brane tiling, denoted as TN(r,s)

Summary

Introduction

Using the constructions of the previous section, we focus on a certain two-parameter family of period 1 quivers These quivers correspond to the Gale-Robinson sequence [Gal91] and were studied, implicitly, in work by Bousquet-Melou, Propp, and West [BPW09]. As explained in Example 8.7 of [FM11], for each triple of positive integers (r, s, N ) with r < s ≤ N/2, there is a unique period 1 quiver whose mutations yield the sequence of xn’s satisfying recurrence (1). Definition 9 (The Gale-Robinson Quiver) For 1 ≤ r ≤ s < N/2, we let Q(Nr,s) denote the quiver constructed by the following four step process, starting with the edge-less quiver on N vertices: 1. In the quiver gauge theory and brane tiling literature, Zn denotes a Pyramid Partition Function (cluster variable) associated to a certain cascade of Seiberg dualities (mutation sequence). Note: When drawing brane tilings or their subgraphs, we will depict hexagonal faces as horizontal rectangles of height one and width two

From Brane Tilings to Pinecones

Principal Coefficients and Combinatorial Formulas

For any of the open faces

Further Topics