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Two enriched poset polytopes

Soichi Okada, Akiyoshi Tsuchiya

Abstract

Stanley introduced and studied two lattice polytopes, the order polytope and chain polytope, associated to a finite poset. Recently Ohsugi and Tsuchiya introduce an enriched version of them, called the enriched order polytope and enriched chain polytope. In this paper, we give a piecewise-linear bijection between these enriched poset polytopes, which is an enriched analogue of Stanley's transfer map and bijectively proves that they have the same Ehrhart polynomials. Also we construct explicitly unimodular triangulations of two enriched poset polytopes.

Two enriched poset polytopes

Abstract

Stanley introduced and studied two lattice polytopes, the order polytope and chain polytope, associated to a finite poset. Recently Ohsugi and Tsuchiya introduce an enriched version of them, called the enriched order polytope and enriched chain polytope. In this paper, we give a piecewise-linear bijection between these enriched poset polytopes, which is an enriched analogue of Stanley's transfer map and bijectively proves that they have the same Ehrhart polynomials. Also we construct explicitly unimodular triangulations of two enriched poset polytopes.

Paper Structure

This paper contains 14 sections, 32 theorems, 42 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Enriched transfer map
  3. Notations
  4. Defining inequalities for enriched poset polytopes
  5. Proof of Theorem \ref{['thm:e-transfer']}
  6. Left enriched $P$-partitions
  7. Vertices of enriched poset polytopes
  8. Triangulations
  9. Poset structure on $\mathcal{F}^{(e)}(P)$
  10. Triangulation of $\mathcal{C}^{(e)}(P)$
  11. Triangulation of $\mathcal{O}^{(e)}(P)$.
  12. Identification with Ohsugi--Tsuchiya's triangulations
  13. Triangulation of enriched order polytope
  14. Triangulation of enriched chain polytope

Key Result

Theorem 1.1

(Stanley Stanley1986) We define a piecewise-linear map $\Phi : \mathbb{R}^P \to \mathbb{R}^P$, called the transfer map, by for $f \in \mathbb{R}^P$ and $v \in P$. Then $\Phi$ induces a continuous bijection from $\mathcal{O}(P)$ to $\mathcal{C}(P)$. In particular, $\Phi$ provides a bijection between $m \mathcal{O}(P) \cap \mathbb{Z}^P$ and $m \mathcal{C}(P) \cap \mathbb{Z}^P$ for any nonnegative i

Figures (3)

  • Figure 1: Hasse diagram of $(\mathcal{F}^{(e)}(\Lambda),\preceq)$
  • Figure 2: Hasse diagram of $(\mathcal{A}^{(e)}(\Lambda),\preceq)$
  • Figure 3: Hasse diagram of $(\mathcal{F}^{(e)}(\Lambda), \ge)$

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • Example 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 28 more