Acyclic systems of permutations and fine mixed subdivisions of simplices

TL;DR

This paper investigates the relationship between acyclic permutation systems and fine mixed subdivisions of simplices, proving key conjectures for the case n=3 and establishing foundational links between permutation systems and simplex collections.

Contribution

It proves that permutation systems determine simplex collections and confirms the Acyclic System and Spread Out Simplices Conjectures for n=3 in any dimension.

Findings

Permutation systems determine the collection of simplices.

Proof of the Acyclic System Conjecture.

Validation of both conjectures for n=3.

Abstract

A fine mixed subdivision of a (d-1)-simplex T of size n gives rise to a system of ${d \choose 2}$ permutations of [n] on the edges of T, and to a collection of n unit (d-1)-simplices inside T. Which systems of permutations and which collections of simplices arise in this way? The Spread Out Simplices Conjecture of Ardila and Billey proposes an answer to the second question. We propose and give evidence for an answer to the first question, the Acyclic System Conjecture. We prove that the system of permutations of T determines the collection of simplices of T. This establishes the Acyclic System Conjecture as a first step towards proving the Spread Out Simplices Conjecture. We use this approach to prove both conjectures for n=3 in arbitrary dimension.