Some acyclic systems of permutations are not realizable by triangulations of a product of simplices

Abstract

The acyclic system conjecture of Ardila and Ceballos can be interpreted as saying the following: "Every triangulation of the 3-skeleton of a product of two simplices can be extended to a triangulation of the whole product". We show a counter-example to this. Motivation for this conjecture comes from a related conjecture, the "spread-out simplices" conjecture of Ardila and Billey. We give some necessary conditions that counter-examples to this second conjecture (if they exist) must satisfy.

arXiv comment12 pages, 7 figures. Version 2 corrects a (significant) typo in v1: Theorem 2.9 had the same statement as Conjecture 2.7, while it was meant to be only the d=2 case of the conjecture