Summary: Acyclic systems of permutations and
fine mixed subdivisions of simplices
A fine mixed subdivision of a d-simplex T of size n gives rise to a system of d+1
permutations of [n] on the edges of T, and to a collection of n unit d-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.
The fine mixed subdivisions of a dilated simplex arise in numerous contexts, and possess a
remarkable combinatorial structure, which has been the subject of great attention recently.
The goal of this paper is to prove several structural results about these subdivisions.