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Gale-Robinson Quivers and Principal Coefficients

Qiyue Chen, Gregg Musiker

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-Mélou-Propp-West, Speyer, Vichitkunakorn, and of Eager-Franco.

Gale-Robinson Quivers and Principal Coefficients

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-Mélou-Propp-West, Speyer, Vichitkunakorn, and of Eager-Franco.
Paper Structure (16 sections, 27 theorems, 44 equations, 21 figures, 1 table)

This paper contains 16 sections, 27 theorems, 44 equations, 21 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries: Periodic Quivers and Cluster Mutation
  3. Gale-Robinson Sequences
  4. From Gale-Robinson Quivers to Brane Tilings
  5. From Brane Tilings to Pinecones
  6. Main Results
  7. Step 1: The recurrence with principal coefficients
  8. Step 2: Counting faces in horizontal strips of pinecones
  9. Step 3: Kuo condensation for pinecones with weights and heights
  10. Kuo's graphical condensation
  11. Pinecones and their core
  12. Kuo's condensation for $x$-variables
  13. Kuo's condensation for $y$-variables
  14. The poset structure on perfect matchings
  15. Proof of Lemma \ref{['lem:superimposition-difference']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\widehat{\mathcal{A}}_{Q_N^{(r,s)}} \subset \mathbb{Q}[y_1,y_2, \dots, y_N][x_1^\pm,x_2^\pm, \dots, x_N^\pm]$ denote the cluster algebra with principal coefficients associated to the Gale-Robinson quiver of type $(r,s,N)$. For $n \in \{N+1,N+2,\dots\}$, define the cluster variables $\widehat{x_ where $x(M)$, $y(M)$ are the weights and heights of perfect matching $M$, respectively.

Figures (21)

  • Figure 1: The Gale-Robinson Quiver $Q^{(2,3)}_7$ as a sum of the primitive period $1$ quivers $-P_7^{(2)}$, $+P_7^{(3)}$, and $-P_{\{4,5,6\}}^{(1)}$.
  • Figure 2: The four possible local configurations in $\widetilde{Q}_N^{(r,s)}$.
  • Figure 3: The unfolded quiver ${\widetilde{Q}^{(2,3)}_7}$ and brane tiling $\mathcal{T}_7^{(2,3)}$.
  • Figure 4: Recovering the Gale-Robinson subgraph for $(r,s,N) = (2,3,7)$ from the associated Aztec Diamond for $10 \leq n \leq 15$. Called the core of a pinecone in ProppMelouWest2009. Compare with Example \ref{['Ex:237']} and $\mathcal{T}_7^{(2,3)}$ of Figure \ref{['Fig:237-BT']}.
  • Figure 5: (Left): Del-Pezzo $2$ quiver $\overline{Q_5^{(1,2)}}$; (Right): Psuedo-Del-Pezzo $2$ quiver $Q_5^{(1,2)}$.
  • ...and 16 more figures

Theorems & Definitions (86)

  • Theorem 1.1
  • Definition 2.1: Quiver Mutation
  • Definition 2.2: Cluster Mutation
  • Definition 2.3: Cluster variables and algebras
  • Definition 2.4: Periodic Quiver
  • Definition 2.5: Quiver with Principal Coefficients
  • Remark 2.6: $\bf{c}$-vectors
  • Definition 3.1: Gale-Robinson Sequences
  • Definition 3.2: The Gale-Robinson Quiver
  • Remark 3.3
  • ...and 76 more