Discrete & Computational Geometry

, Volume 49, Issue 3, pp 485–510

Acyclic Systems of Permutations and Fine Mixed Subdivisions of Simplices


DOI: 10.1007/s00454-013-9485-1

Cite this article as:
Ardila, F. & Ceballos, C. Discrete Comput Geom (2013) 49: 485. doi:10.1007/s00454-013-9485-1


A fine mixed subdivision of a \((d-1)\)-simplex \(T\) of size \(n\) gives rise to a system of \({d \atopwithdelims ()2}\) 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.


Fine mixed subdivisions Triangulations Product of simplices Tropical geometry 

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsSan Francisco State UniversitySan FranciscoUSA
  2. 2.Institut für MathematikFreie Universität BerlinBerlinGermany

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