Discrete & Computational Geometry

, Volume 49, Issue 3, pp 485–510 | Cite as

Acyclic Systems of Permutations and Fine Mixed Subdivisions of Simplices



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 


  1. 1.
    Ardila, F., Beck, M., Hosten, S., Pfeifle, J., Seashore, K.: Root polytopes and growth series of root lattices. SIAM J. Discrete Math. 25, 360–378 (2011)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Ardila, F., Billey, S.: Flag arrangements and triangulations of products of simplices. Adv. Math. 214, 495–524 (2007)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Ardila, F., Develin, M.: Tropical hyperplane arrangements and oriented matroids. Math. Z. 262, 795–816 (2009)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Babson, E., Billera, L.: The geometry of products of minors. Discrete Comput. Geom. 20, 231–249 (1998)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bayer, M.: Equidecomposable and weakly neighborly polytopes. Israel J. Math. 81, 301–320 (1993)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Billey, S., Vakil, R.: Intersections of Schubert varieties and other permutation array schemes. In: IMA Volumes in Mathematics and its Applications, vol. 146: Algorithms in Algebraic Geometry, Springer, New York, pp. 21–54 (2008)Google Scholar
  7. 7.
    De Loera, J.A., Rambau, J., Francisco, S.: Triangulations: structures for algorithms and applications. In: Algorithms and Computation in Mathematics, vol. 25. Springer, Berlin (2010)Google Scholar
  8. 8.
    Develin, M., Sturmfels, B.: Tropical convexity. Doc. Math. 9, 1–27 (2004)MathSciNetMATHGoogle Scholar
  9. 9.
    Gelfand, I.M., Kapranov, M., Zelevinsky, A.: Discriminants, resultants and multidimensional determinants. Birkhäuser, Boston (1994)MATHCrossRefGoogle Scholar
  10. 10.
    Haiman, M.: A simple and relatively efficient triangulation of the $n$-cube. Discrete Comput. Geom. 6, 287–289 (1991)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Herrmann, S., Jensen, A., Joswig, M., Sturmfels, B.: How to draw tropical planes. Electron. J. Combin. 16 (2009) (Special volume in honor of Anders Björner, Research Paper 6, 26 pp)Google Scholar
  12. 12.
    Huber, B., Rambau, J., Santos, F.: The Cayley trick, lifting subdivisions and the Bohne–Dress theorem on zonotopal tilings. J. Eur. Math. Soc. (JEMS) 2(2), 179–198 (2000)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Luby, M., Randall, D., Sinclair, A.: Markov chain algorithms for planar lattice structures. In: IEEE Symposium on Foundations of Computer Science, pp. 150–159 (1995)Google Scholar
  14. 14.
    Oh, S., Yoo, H.: Triangulations of $\Delta _{n-1} \times \Delta _{d-1}$ and tropical oriented matroids. In: Discrete Mathematics and Theoretical Computer Science Proceedings, FPSAC 2011, pp. 717–728Google Scholar
  15. 15.
    Orden, D., Santos, F.: Asymptotically efficient triangulations of the $d$-cube. Discrete Comput. Geom. 30, 509–528 (2003)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Oxley, J.: Matroid Theory. Oxford University Press, New York (1992)MATHGoogle Scholar
  17. 17.
    Santos, F.: A point configuration whose space of triangulations is disconnected. J. Am. Math. Soc. 13, 611–637 (2000)MATHCrossRefGoogle Scholar
  18. 18.
    Santos, F.: Non-connected toric Hilbert schemes. Math. Ann. 332, 645–665 (2005)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Santos, F.: Some acyclic systems of permutations are not realizable by triangulations of a product of simplices. In: Proceedings of the CIEM Workshop in Tropical Geometry, Contemporary Math, AMS (to appear)Google Scholar
  20. 20.
    Santos, F.: The Cayley trick and triangulations of products of simplices. In: Integer Points in Polyhedra—Geometry, Number Theory, Algebra, Optimization. Contemporary Mathematics, vol. 374. American Mathematical Society, Providence, RI, pp. 151–177 (2005)Google Scholar
  21. 21.
    Sturmfels, B.: Grobner Bases and Convex Polytopes. University Lectures Series, No 8, American Mathematical Society, Providence, RI (1996)Google Scholar

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

Personalised recommendations