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ABHY Associahedra and Newton polytopes of $F$-polynomials for finite type cluster algebras

Véronique Bazier-Matte, Nathan Chapelier-Laget, Guillaume Douville, Kaveh Mousavand, Hugh Thomas, Emine Yıldırım

Abstract

A new construction of the associahedron was recently given by Arkani-Hamed, Bai, He, and Yan in connection with the physics of scattering amplitudes. We show that their construction (suitably understood) can be applied to construct generalized associahedra of any simply-laced Dynkin type. Unexpectedly, we also show that this same construction produces Newton polytopes for all the $F$-polynomials of the corresponding cluster algebras. In addition, we show that the toric variety associated to the g-vector fan has the property that its nef cone is simplicial.

ABHY Associahedra and Newton polytopes of $F$-polynomials for finite type cluster algebras

Abstract

A new construction of the associahedron was recently given by Arkani-Hamed, Bai, He, and Yan in connection with the physics of scattering amplitudes. We show that their construction (suitably understood) can be applied to construct generalized associahedra of any simply-laced Dynkin type. Unexpectedly, we also show that this same construction produces Newton polytopes for all the -polynomials of the corresponding cluster algebras. In addition, we show that the toric variety associated to the g-vector fan has the property that its nef cone is simplicial.

Paper Structure

This paper contains 12 sections, 19 theorems, 40 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Construction
  3. Proof of correctness
  4. Background on cluster algebras
  5. Submodule polytopes and torsion classes
  6. Newton polytopes of $F$-polynomials
  7. The use of principal coefficients
  8. Universal coefficients
  9. The nef cone of the toric variety associated to the $g$-vector fan is simplicial
  10. Brief reminder on toric varieties
  11. Combinatorics of $\operatorname{Pic}(X_\Sigma)$
  12. Proof of Theorem \ref{['simp']}

Key Result

Theorem 1

Figures (1)

  • Figure 1: Associahedron corresponding to $1 \rightarrow 2 \leftarrow3$.

Theorems & Definitions (49)

  • Example 1
  • Example 2
  • Theorem 1
  • Example 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 39 more