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A canonical realization of the alt $ν$-associahedron

Cesar Ceballos

Abstract

Given a lattice path $ν$, the alt $ν$-Tamari lattice is a partial order recently introduced by Ceballos and Chenevière, which generalizes the $ν$-Tamari lattice and the $ν$-Dyck lattice. All these posets are defined on the set of lattice paths that lie weakly above $ν$, and posses a rich combinatorial structure. In this paper, we study the geometric structure of these posets. We show that their Hasse diagram is the edge graph of a polytopal complex induced by a tropical hyperplane arrangement, which we call the alt $ν$-associahedron. This generalizes the realization of $ν$-associahedra by Ceballos, Padrol and Sarmiento. Our approach leads to an elegant construction, in terms of areas below lattice paths, which we call the canonical realization. Surprisingly, in the case of the classical associahedron, our canonical realization magically recovers Loday's ubiquitous realization, via a simple affine transformation.

A canonical realization of the alt $ν$-associahedron

Abstract

Given a lattice path , the alt -Tamari lattice is a partial order recently introduced by Ceballos and Chenevière, which generalizes the -Tamari lattice and the -Dyck lattice. All these posets are defined on the set of lattice paths that lie weakly above , and posses a rich combinatorial structure. In this paper, we study the geometric structure of these posets. We show that their Hasse diagram is the edge graph of a polytopal complex induced by a tropical hyperplane arrangement, which we call the alt -associahedron. This generalizes the realization of -associahedra by Ceballos, Padrol and Sarmiento. Our approach leads to an elegant construction, in terms of areas below lattice paths, which we call the canonical realization. Surprisingly, in the case of the classical associahedron, our canonical realization magically recovers Loday's ubiquitous realization, via a simple affine transformation.
Paper Structure (14 sections, 20 theorems, 42 equations, 33 figures)

This paper contains 14 sections, 20 theorems, 42 equations, 33 figures.

Table of Contents

  1. Introduction
  2. The alt $\nu$-Tamari lattice
  3. The alt $\nu$-Tamari lattice on paths
  4. The $U$-Tamari lattice on trees
  5. The isomorphism between both lattices
  6. The $U$-Tamari complex
  7. Geometric realizations
  8. Triangulation of the polytope $S_{U}$
  9. Fine mixed subdivision of the polytope $\mathcal{P}_{U}$
  10. Tropical hyperplane arrangement realization
  11. The $U$-associahedron
  12. Defining inequalities
  13. Coordinates of the vertices
  14. A canonical realization

Key Result

Theorem 1.1

Let $\nu$ be a lattice path from $(0,0)$ to (a,b), and $\delta$ be an increment vector with respect to $\nu$. The Hasse diagram of the alt $\nu$-Tamari lattice $\operatorname{Tam}_{\nu}(\delta)$ can be realized geometrically as:

Figures (33)

  • Figure 1: Example of Loday's coordinates of two plane binary trees.
  • Figure 2: Loday's 3-dimensional associahedron.
  • Figure 3: Example of coordinates of two plane binary trees in our canonical realization.
  • Figure 4: Our canonical realization of the 3-dimensional associahedron.
  • Figure 5: The $\nu$-Tamari lattice and $\nu$-Dyck lattice for $\nu=ENEENN=(1,2,0,0)$. They are the alt $\nu$-Tamari lattices $\operatorname{Tam}_{\nu}(\delta)$ for $\delta=(2,0,0)$ and $\delta=(0,0,0)$, respectively.
  • ...and 28 more figures

Theorems & Definitions (59)

  • Theorem 1.1: Theorem \ref{['thm_noncrossing_triangulation']}, Corollary \ref{['cor_fine_mixed_subdivision']}, and Definition \ref{['def_U_associhedron']}/Theorem \ref{['def_thm_U_associahedron']}
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3: ceballos_cheneviere_linear_2023
  • Definition 2.4
  • Definition 2.5
  • Example 2.6
  • Theorem 2.7: ceballos_cheneviere_linear_2023
  • Remark 2.8
  • Remark 2.9
  • ...and 49 more