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

TL;DR

This paper introduces a toric analogue of Greene's rational function for posets, explores its properties, and connects it to scattering amplitude relations, providing new tools and insights in combinatorics and mathematical physics.

Contribution

It defines a new toric version of Greene's rational function, analyzes its properties, and links it to Kleiss-Kuijf relations in scattering amplitudes.

Findings

Toric analogue of Greene's rational function is well-defined and studied.

Connections established between toric posets and scattering amplitude relations.

An algorithm for computing toric total extensions is provided.

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.