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Deformed graphical zonotopes

Arnau Padrol, Vincent Pilaud, Germain Poullot

TL;DR

This work provides irredundant descriptions of the deformation cones for graphical zonotopes, connecting deformations of deformed permutahedra to the combinatorics of the underlying graph. By refining the braid fan and exploiting a simplicial refinement, the authors obtain explicit linear-span bases given by Minkowski summands corresponding to nonempty induced cliques and a precise facet description tied to edge/non-edge relations. A key structural result is that the deformation cone is simplicial precisely when the graph is triangle-free, in which case every deformation is a sum of edge-based generators $\Delta_e$, otherwise the cone is non-simplicial and richer in structure. These results situate graphical deformation cones within the larger submodular/cone framework and have potential implications for canonical polytope realizations in geometric combinatorics and positive geometries such as amplituhedra.

Abstract

We study deformations of graphical zonotopes. Deformations of the classical permutahedron (which is the graphical zonotope of the complete graph) have been intensively studied in recent years under the name of generalized permutahedra. We provide an irredundant description of the deformation cone of the graphical zonotope associated to a graph $G$, consisting of independent equations defining its linear span (in terms of non-cliques of $G$) and of the inequalities defining its facets (in terms of common neighbors of neighbors in $G$). In particular, we deduce that the faces of the standard simplex corresponding to induced cliques in $G$ form a linear basis of the deformation cone, and that the deformation cone is simplicial if and only if $G$ is triangle-free.

Deformed graphical zonotopes

TL;DR

This work provides irredundant descriptions of the deformation cones for graphical zonotopes, connecting deformations of deformed permutahedra to the combinatorics of the underlying graph. By refining the braid fan and exploiting a simplicial refinement, the authors obtain explicit linear-span bases given by Minkowski summands corresponding to nonempty induced cliques and a precise facet description tied to edge/non-edge relations. A key structural result is that the deformation cone is simplicial precisely when the graph is triangle-free, in which case every deformation is a sum of edge-based generators , otherwise the cone is non-simplicial and richer in structure. These results situate graphical deformation cones within the larger submodular/cone framework and have potential implications for canonical polytope realizations in geometric combinatorics and positive geometries such as amplituhedra.

Abstract

We study deformations of graphical zonotopes. Deformations of the classical permutahedron (which is the graphical zonotope of the complete graph) have been intensively studied in recent years under the name of generalized permutahedra. We provide an irredundant description of the deformation cone of the graphical zonotope associated to a graph , consisting of independent equations defining its linear span (in terms of non-cliques of ) and of the inequalities defining its facets (in terms of common neighbors of neighbors in ). In particular, we deduce that the faces of the standard simplex corresponding to induced cliques in form a linear basis of the deformation cone, and that the deformation cone is simplicial if and only if is triangle-free.
Paper Structure (9 sections, 12 theorems, 27 equations, 3 figures)

This paper contains 9 sections, 12 theorems, 27 equations, 3 figures.

Table of Contents

  1. Preliminaries
  2. Fans and polytopes
  3. Deformation cones
  4. Graphical zonotopes
  5. Graphical deformation cones
  6. A first polyhedral description
  7. The linear span of graphical deformation cones
  8. The facets of graphical deformation cones
  9. Simplicial graphical deformation cones

Key Result

Proposition 1.1

Let $\mathsf{P} \subset \mathbb{R}^d$ be a simple polytope with simplicial normal fan $\Fan$ supported on the rays ${\boldsymbol{S}}$. Then the deformation cone $\mathbb{DC}(\mathsf{P})$ is the set of polytopes $\mathsf{P}_{\boldsymbol{h}}$ for all ${\boldsymbol{h}}$ in the cone of $\mathbb{R}^{\bol for all adjacent maximal cones $\mathbb{R}_{\ge0}{\boldsymbol{R}}$ and $\mathbb{R}_{\ge0}{\boldsymb

Figures (3)

  • Figure 1: The fan $\sbraid[123]$ intersected with the unit sphere. (For brevity, here and in the labels we write $123$ to denote the set $\{1,2,3\}$, and so on.) The braid fan $\braid[123]$ is the Cartesian product of a regular hexagonal fan with a line. To obtain $\sbraid[123]$, each maximal cell is divided into two simplicial cells, one containing ${\boldsymbol{\iota}}_{\varnothing}$ and one containing ${\boldsymbol{\iota}}_{123}$.
  • Figure 2: A $3$-dimensional affine section of the deformation cone $\mathbb{DC}(\gZono[K_3])$ for the triangle $K_3$. The deformations of $\gZono[K_3]$ corresponding to some of the points of $\mathbb{DC}(\gZono[K_3])$ are depicted.
  • Figure 3: A $3$-dimensional affine section of the deformation cone $\mathbb{DC}(\gZono[C_4])$ for the $4$-cycle $C_4$. The deformations of $\gZono[C_4]$ corresponding to some of the points of $\mathbb{DC}(\gZono[C_4])$ are depicted.

Theorems & Definitions (20)

  • Proposition 1.1: GelfandKapranovZelevinskyChapotonFominZelevinsky
  • Proposition 1.2
  • Proposition 1.3
  • Example 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • ...and 10 more