[Paper Review] A realization of poset associahedra

Andrew Sack|arXiv (Cornell University)|Jan 26, 2023

Advanced Combinatorial Mathematics4 citations

TL;DR

This paper provides a new explicit realization of Galashin's poset associahedra and affine poset cyclohedra as convex polytopes in ℝ^P by defining them as intersections of half-spaces using a linear form α_τ that approximates the diameter of tubes in the poset. The key contribution is a geometric construction that realizes these polytopes via inequalities derived from order-preserving maps and compactification principles, offering a direct, computable description of their face structure.

ABSTRACT

Given any connected poset $P$, we give a simple realization of Galashin's poset associahedron $\mathscr{A}(P)$ as a convex polytope in $\mathbb{R}^P.$ The realization is inspired by the description of $\mathscr{A}(P)$ as a compactification of the configuration space of order-preserving maps $P o \mathbb{R}.$ In addition, we give an analogous realization for Galashin's affine poset cyclohedra.

Motivation & Objective

• To provide an explicit geometric realization of Galashin's abstractly constructed poset associahedra as convex polytopes in ℝ^P.
• To extend this realization to Galashin's affine poset cyclohedra using analogous constructions.
• To resolve the lack of explicit realizations for poset associahedra despite their combinatorial significance and connection to configuration space compactifications.
• To offer a computationally accessible description of the face lattice via inequalities derived from tube diameter approximations.

Proposed method

• Define a linear functional α_τ on ℝ^P_Σ=0 (sum-zero functions) that sums over covering relations in a tube τ, approximating the diameter of τ.
• Construct half-spaces h_τ = {p ∈ ℝ^P_Σ=0 | α_τ(p) ≥ n^{2|τ|}} and hyperplanes H_τ = {p | α_τ(p) = n^{2|τ|}}.
• Realize the poset associahedron 𝒜(P) as the intersection of H_P with all h_τ over proper tubes τ ⊆ P.
• Use the acyclic tubing condition and periodicity to extend the construction to affine poset cyclohedra.
• Prove that vertices correspond to maximal tubings by showing that only maximal tubings satisfy equality in all defining inequalities.
• Leverage the compactification of order-preserving maps P → ℝ as motivation for the choice of the threshold n^{2|τ|} in the inequalities.

Experimental results

Research questions

• RQ1 Can a simple, explicit realization of the poset associahedron be constructed as a convex polytope in ℝ^P, independent of stellar subdivisions?
• RQ2 How can the face lattice of the poset associahedron be geometrically realized via inequalities on the ambient space?
• RQ3 Can the same approach be extended to Galashin’s affine poset cyclohedra, which generalize the finite case?
• RQ4 What is the role of the linear form α_τ in approximating tube diameter and ensuring correct face incidence?
• RQ5 Is there a natural choice of α_τ that preserves the combinatorial structure while enabling computational access to the polytope?

Key findings

• The poset associahedron 𝒜(P) is realized as the intersection of the hyperplane H_P with all half-spaces h_τ for proper tubes τ ⊆ P.
• Vertices of 𝒜(P) correspond exactly to maximal tubings of P, as these are the only points satisfying equality in all defining inequalities.
• The construction uses a threshold of n^{2|τ|} for α_τ(p), which ensures that only maximal tubings lie on the boundary of all h_τ.
• For affine poset cyclohedra, the realization extends via n-periodic tubings and the same inequality system, preserving the acyclic tubing condition.
• The realization is compatible with the compactification of order-preserving maps P → ℝ, providing a geometric interpretation of the polytope as a compactified configuration space.
• Alternative choices for α_τ (e.g., summing over all covering pairs or spanning trees) yield isomorphic realizations, suggesting robustness in the construction.