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Associahedra as moment polytopes

Michael Gekhtman, Hugh Thomas

Abstract

Generalized associahedra are a well-studied family of polytopes associated to a finite-type cluster algebra and choice of starting cluster. We show that the generalized associahedra constructed by Padrol, Palu, Pilaud, and Plamondon, building on ideas from Arkani-Hamed, Bai, He, and Yan, can be naturally viewed as moment polytopes for an open patch of the quotient of the cluster A-variety with universal coefficients by its maximal natural torus action. We prove our result by showing that the construction of Padrol, Palu, Pilaud, and Plamondon can be understood on the basis of the way that moment polytopes behave under symplectic reduction.

Associahedra as moment polytopes

Abstract

Generalized associahedra are a well-studied family of polytopes associated to a finite-type cluster algebra and choice of starting cluster. We show that the generalized associahedra constructed by Padrol, Palu, Pilaud, and Plamondon, building on ideas from Arkani-Hamed, Bai, He, and Yan, can be naturally viewed as moment polytopes for an open patch of the quotient of the cluster A-variety with universal coefficients by its maximal natural torus action. We prove our result by showing that the construction of Padrol, Palu, Pilaud, and Plamondon can be understood on the basis of the way that moment polytopes behave under symplectic reduction.
Paper Structure (13 sections, 2 theorems, 17 equations)

This paper contains 13 sections, 2 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Other approaches to viewing associahedra as moment polytopes
  3. Background
  4. Cluster algebras
  5. Universal cluster algebras
  6. An example
  7. Associahedra
  8. Degenerate associahedra as Newton polytopes
  9. Symplectic reduction
  10. Cluster varieties
  11. Proof of the main result
  12. Symplectic reduction of a patch of $\mathcal{M}(B^\textit{univ})$
  13. An example

Key Result

Proposition 1

The moment map image of $X_\mathit{red}$ with respect to the action of $R_1$ is given by $(i^*)^{-1}(\hat{c}) \cap \mu_R(X)$, where we identify $(i^*)^{-1}(\hat{c})$ with $\mathfrak r^*_1$.

Theorems & Definitions (4)

  • Proposition 1
  • Theorem 1
  • Remark 1
  • Remark 2