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A Toric Analogue for Greene's Rational Function of a Poset

Elise Catania

TL;DR

This work extends Greene's rational function from ordinary posets to toric posets by defining a toric analogue $\Psi^{[Q]}_{\mathrm{tor}}(\mathbf{x})$ and the set of toric total extensions, $\mathcal{L}_{\mathrm{tor}}([Q])$. It establishes a robust combinatorial-dgeometric framework: a bijection between toric chambers and acyclic quivers modulo sink/source flips, a toric transitive closure/toric Hasse diagram framework, and a recursion to compute toric total extensions, along with a vanishing criterion and a denominator description echoing the classical case. The paper also uncovers connections to scattering amplitudes via Kleiss–Kuijf relations and Parke–Taylor factors, and proves counting toric total extensions is $\#P$-complete, highlighting intrinsic computational complexity. Collectively, the results fuse poset theory, toric geometry, and amplitude combinatorics, providing both theoretical insight and algorithmic tools for toric total extensions.

Abstract

Given a finite poset, Greene introduced a rational function obtained by summing certain rational functions over the linear extensions of the poset. This function has interesting interpretations, and for certain families of posets, it simplifies surprisingly. In particular, Greene evaluated this rational function for strongly planar posets in his work on the Murnaghan-Nakayama formula. In 2012, Develin, Macauley, and Reiner introduced toric posets, which combinatorially are equivalence classes of posets (or rather acyclic quivers) under the operation of flipping maximum elements into minimum elements and vice versa. In this work, we introduce a toric analogue of Greene's rational function for toric posets, and study its properties. In addition, we use toric posets to show that the Kleiss-Kuijf relations, which appear in scattering amplitudes, are equivalent to a specific instance of Greene's evaluation of his rational function for strongly planar posets. Also in this work, we give an algorithm for finding the set of toric total extensions of a toric poset.

A Toric Analogue for Greene's Rational Function of a Poset

TL;DR

This work extends Greene's rational function from ordinary posets to toric posets by defining a toric analogue and the set of toric total extensions, . It establishes a robust combinatorial-dgeometric framework: a bijection between toric chambers and acyclic quivers modulo sink/source flips, a toric transitive closure/toric Hasse diagram framework, and a recursion to compute toric total extensions, along with a vanishing criterion and a denominator description echoing the classical case. The paper also uncovers connections to scattering amplitudes via Kleiss–Kuijf relations and Parke–Taylor factors, and proves counting toric total extensions is -complete, highlighting intrinsic computational complexity. Collectively, the results fuse poset theory, toric geometry, and amplitude combinatorics, providing both theoretical insight and algorithmic tools for toric total extensions.

Abstract

Given a finite poset, Greene introduced a rational function obtained by summing certain rational functions over the linear extensions of the poset. This function has interesting interpretations, and for certain families of posets, it simplifies surprisingly. In particular, Greene evaluated this rational function for strongly planar posets in his work on the Murnaghan-Nakayama formula. In 2012, Develin, Macauley, and Reiner introduced toric posets, which combinatorially are equivalence classes of posets (or rather acyclic quivers) under the operation of flipping maximum elements into minimum elements and vice versa. In this work, we introduce a toric analogue of Greene's rational function for toric posets, and study its properties. In addition, we use toric posets to show that the Kleiss-Kuijf relations, which appear in scattering amplitudes, are equivalent to a specific instance of Greene's evaluation of his rational function for strongly planar posets. Also in this work, we give an algorithm for finding the set of toric total extensions of a toric poset.
Paper Structure (3 sections, 6 theorems, 20 equations, 1 figure, 1 table)

This paper contains 3 sections, 6 theorems, 20 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Posets and Graphic Hyperplane Arrangements
  3. Toric Posets

Key Result

Theorem 1.5

Let $[Q]$ be a toric poset, and let $G$ be the underlying undirected graph of $[Q]$. If $G$ is either disconnected with at least three vertices or has a cut vertex, then $\Psi_{\rm{tor}}^{[Q]}(\mathbf{x})=0$.

Figures (1)

  • Figure 1: One representative of $[Q]$ from Corollary \ref{['cor: Kleiss-Kuijf relations']}.

Theorems & Definitions (19)

  • Example 1.1
  • Example 1.2
  • Definition 1.3
  • Example 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Lemma 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 9 more