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Polytopal realizations of non-crystallographic associahedra

Anna Felikson, Pavel Tumarkin, Emine Yildirim

TL;DR

The paper addresses realizing generalized associahedra for non-simply-laced and non-crystallographic root systems by embedding them as planar sections of simply-laced realizations constructed via representation theory. It introduces a folding-based plane $\Pi$ and proves that $\Pi\cap \mathbb{A}_Q$ yields the generalized associahedron $\mathbb{A}_{\Delta'}$, with its normal fan being the $\mathbf{g}$-vector fan, obtainable as an orthogonal projection of the $\mathbf{g}$-vectors from the unfolding $Q$ of type $\Delta$. The key contribution is establishing a precise, polyhedral realization for non-simply-laced and non-crystallographic cases, generalizing AHL in crystallographic settings and using weighted unfoldings and tropical duality to handle non-crystallographic types. This provides a unified, constructive approach to polytopal realizations across finite types, with potential implications for cluster algebras and representation-theoretic interpretations of these polytopes.

Abstract

We use the folding technique to show that generalized associahedra for non-simply-laced root systems (including non-crystallographic ones) can be obtained as sections of simply-laced generalized associahedra constructed by Bazier-Matte, Chapelier-Laget, Douville, Mousavand, Thomas and Yildirim.

Polytopal realizations of non-crystallographic associahedra

TL;DR

The paper addresses realizing generalized associahedra for non-simply-laced and non-crystallographic root systems by embedding them as planar sections of simply-laced realizations constructed via representation theory. It introduces a folding-based plane and proves that yields the generalized associahedron , with its normal fan being the -vector fan, obtainable as an orthogonal projection of the -vectors from the unfolding of type . The key contribution is establishing a precise, polyhedral realization for non-simply-laced and non-crystallographic cases, generalizing AHL in crystallographic settings and using weighted unfoldings and tropical duality to handle non-crystallographic types. This provides a unified, constructive approach to polytopal realizations across finite types, with potential implications for cluster algebras and representation-theoretic interpretations of these polytopes.

Abstract

We use the folding technique to show that generalized associahedra for non-simply-laced root systems (including non-crystallographic ones) can be obtained as sections of simply-laced generalized associahedra constructed by Bazier-Matte, Chapelier-Laget, Douville, Mousavand, Thomas and Yildirim.
Paper Structure (7 sections, 7 theorems, 4 equations, 2 figures, 1 table)

This paper contains 7 sections, 7 theorems, 4 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Simply-laced generalized associahedra
  4. ${\mathbf g}$-vectors
  5. Unfoldings
  6. Construction and main result
  7. Proof of the main result

Key Result

Theorem 1.1

The section of ${\mathbb A}_\Delta$ by $\Pi$ is a realization of the generalized associahedron ${\mathbb A}_{\Delta'}$. The normal fan of ${\mathbb A}_{\Delta'}$ is the corresponding ${\mathbf g}$-vector fan, and it can be obtained as an orhogonal projection of the normal fan of ${\mathbb A}_\Delta$

Figures (2)

  • Figure 2.1: Associahedron of type $A_3$, see Example \ref{['cat-a3']}.
  • Figure 3.1: Intersection plane $\Pi$ that yields the associahedron for $C_2$, see Example \ref{['ex-c2']}.

Theorems & Definitions (18)

  • Theorem 1.1: Theorem \ref{['main']}
  • Example 2.1
  • Remark 2.2
  • Definition 3.1
  • Remark 3.2
  • Example 3.3
  • Theorem 3.4
  • Proposition 4.1
  • proof
  • Example 4.2
  • ...and 8 more