Discrete & Computational Geometry

, Volume 54, Issue 1, pp 195–231 | Cite as

Fan Realizations of Type \(A\) Subword Complexes and Multi-associahedra of Rank 3

  • Nantel Bergeron
  • Cesar Ceballos
  • Jean-Philippe Labbé


We present complete simplicial fan realizations of any spherical subword complex of type \(A_n\) for \(n\le 3\). This provides complete simplicial fan realizations of simplicial multi-associahedra \(\varDelta _{2k+4,k}\), whose facets are in correspondence with \(k\)-triangulations of a convex \((2k+4)\)-gon. This solves the first open case of the problem of finding fan realizations where polytopality is not known. We also present fan realizations of two previously unknown cases of subword complexes of type \(A_4\), namely the multi-associahedra \(\varDelta _{9,2}\) and \(\varDelta _{11,3}\).


Simplicial fans Multi-associahedra Subword complex Gale duality Realization space 

Mathematics Subject Classification

52B11 20F55 


  1. 1.
    Altshuler, A., Steinberg, L.: An enumeration of combinatorial 3-manifolds with nine vertices. Discrete Math. 16(2), 91–108 (1976)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Arkani-Hamed, N., Bourjaily, J.L., Cachazo, F., Goncharov, A.B., Postnikov, A., Trnka, J.: Scattering amplitudes and the positive grassmannian. Preprint, arXiv:1212.5605v2, 158 pages (2014)
  3. 3.
    Björner, A., Brenti, F.: Combinatorics of Coxeter Groups. GTM, vol. 231. Springer, Berlin (2005)MATHGoogle Scholar
  4. 4.
    Bokowski, J., Pilaud, V.: On symmetric realizations of the simplicial complex of 3-crossing-free sets of diagonals of the octagon. In: Proceedings of the 21st Canadian Conference on Computational Geometry (CCCG2009), pp. 41–44 (2009). http://cccg.ca/proceedings/2009/cccg09_11
  5. 5.
    Ceballos, C.: On associahedra and related topics. Ph.D. thesis, Freie Universität Berlin, Berlin (2012). http://www.diss.fu-berlin.de/diss/receive/FUDISS_thesis_000000039026
  6. 6.
    Ceballos, C., Ziegler, G.M.: Realizing the associahedron: mysteries and questions. In: Müller-Hoissen, F., Pallo, J.M., Stasheff, J. (eds.) Associahedra, Tamari Lattices and Related Structures. Progress in Mathematics, vol. 299, pp. 119–127. Springer, Basel (2012)CrossRefGoogle Scholar
  7. 7.
    Ceballos, C., Labbé, J.P., Stump, C.: Subword complexes, cluster complexes, and generalized multi-associahedra. J. Algebr. Comb. 39(1), 17–51 (2014)CrossRefMATHGoogle Scholar
  8. 8.
    Ceballos, C., Santos, F., Ziegler, G.M.: Many non-equivalent realizations of the associahedron. Combinatorica. (2014). http://arxiv.org/abs/1109.5544. doi:10.1007/s00493-014-2959-9
  9. 9.
    De Loera, J.A., Rambau, J., Santos, F.: Triangulations, Algorithms and Computation in Mathematics, vol. 25. Springer, Berlin (2010)Google Scholar
  10. 10.
    Elnitsky, S.: Rhombic tilings of polygons and classes of reduced words in Coxeter groups. J. Comb. Theory, Ser. A 77(2), 193–221 (1997)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Felsner, S., Weil, H.: A theorem on higher Bruhat orders. Discrete Comput. Geom. 23(1), 121–127 (2000)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Hohlweg, C., Lange, C.E.M.C.: Realizations of the associahedron and cyclohedron. Discrete Comput. Geom. 37(4), 517–543 (2007)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press, Cambridge (1992)MATHGoogle Scholar
  14. 14.
    Jonsson, J.: Generalized triangulations and diagonal-free subsets of stack polyominoes. J. Comb. Theory, Ser. A 112(1), 117–142 (2005)CrossRefMATHGoogle Scholar
  15. 15.
    Knutson, A., Miller, E.: Subword complexes in Coxeter groups. Adv. Math. 184(1), 161–176 (2004)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Knutson, A., Miller, E.: Gröbner geometry of Schubert polynomials. Ann. Math. 161(3), 1245–1318 (2005)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Manin, Y., Shekhtman, V.: Arrangements of hyperplanes, higher braid groups and higher Bruhat orders. Explicit Universal Deformations of Galois Representations. Advanced Studies in Pure Mathematics, pp. 289–308. Academic Press, Boston (1989)Google Scholar
  18. 18.
    Müller-Hoissen, F., Pallo, J.M., Stasheff, J. (eds.): Associahedra, Tamari Lattices and Related Structures. Progress in Mathematics, vol. 299. Birkhäuser Boston Inc., Boston (2012)MATHGoogle Scholar
  19. 19.
    Pilaud, V., Santos, F.: Multitriangulations as complexes of star polygons. Discrete Comput. Geom. 41(2), 284–317 (2009)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Pilaud, V., Pocchiola, M.: Multitriangulations, pseudotriangulations and primitive sorting networks. Discrete Comput. Geom. 48(1), 142–191 (2012)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Pilaud, V., Santos, F.: The brick polytope of a sorting network. Eur. J. Comb. 33(4), 632–662 (2012)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Pilaud, V., Stump, C.: Brick polytopes of spherical subword complexes: a new approach to generalized associahedra. Adv. Math. 276, 1–61 (2015)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Reiner, V., Roichman, Y.: Diameter of graphs of reduced words and galleries. Trans. Am. Math. Soc. 365(5), 2779–2802 (2013)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Richter-Gebert, J.: Realization Spaces of Polytopes. Springer, Berlin (1996)Google Scholar
  25. 25.
    Rote, G., Santos, F., Streinu, I.: Expansive motions and the polytope of pointed pseudo-triangulations. Discrete and Computational Geometry. Algorithms Combin., pp. 699–736. Springer, Berlin (2003)CrossRefGoogle Scholar
  26. 26.
    Serrano, L., Stump, C.: Maximal fillings of moon polyominoes, simplicial complexes, and Schubert polynomials. Electron. J. Comb. 19(1), P16 (2012) http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i1p16
  27. 27.
    Shapiro, B., Shapiro, M., Vainshtein, A.: Connected components in the intersection of two open opposite Schubert cells in \(SL_n({\mathbb{R}})/B\). Int. Math. Res. Not. 1997(10), 469–493 (1997)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Stein, W.A., et al.: Sage Mathematics Software (Version 6.2). The Sage Development Team (2014). http://www.sagemath.org
  29. 29.
    Stump, C.: A new perspective on \(k\)-triangulations. J. Comb. Theory, Ser. A 118(6), 1794–1800 (2011)Google Scholar
  30. 30.
    Tits, J.: Le problème des mots dans les groupes de Coxeter. Symposia Mathematica (INDAM. Rome, 1967/68), pp. 175–185. Academic Press, London (1969)Google Scholar
  31. 31.
    Ziegler, G.M.: Higher Bruhat orders and cyclic hyperplane arrangements. Topology 32(2), 259–279 (1993)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Nantel Bergeron
    • 1
  • Cesar Ceballos
    • 1
  • Jean-Philippe Labbé
    • 2
  1. 1.Fields InstituteYork UniversityTorontoCanada
  2. 2.Institut für MathematikFreie Universität BerlinBerlinGermany

Personalised recommendations