Advertisement

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
Article

Abstract

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 \(\varDelta _{11,3}\).

Keywords

Simplicial fans Multi-associahedra Subword complex Gale duality Realization space 

Mathematics Subject Classification

52B11 20F55 

Notes

Acknowledgments

The first author was partially supported by NSERC. The second author was supported by the government of Canada through an NSERC Banting Postdoctoral Fellowship. He was also supported by a York University research grant. The third author was supported by a FQRNT Doctoral scholarship and SFB Transregio “Discretization in Geometry and Dynamics” (TRR 109). The authors are grateful to Bruno Benedetti, Frank Lutz, Thomas McConville, and Vic Reiner for important conversations that influenced the results in this paper. They are especially grateful to Darij Grinberg for his important comments about Sect. 3, to Francisco Santos for his polytopal construction in Example 6, and to an anonymous referee for suggesting to include Remark 5 about the sign function for finite Coxeter groups. They are also grateful to Vincent Pilaud and Vic Reiner for their comments on previous versions of this paper, and thank two anonymous referees for their careful reading and suggestions.

References

  1. 1.
    Altshuler, A., Steinberg, L.: An enumeration of combinatorial 3-manifolds with nine vertices. Discrete Math. 16(2), 91–108 (1976)CrossRefzbMATHMathSciNetGoogle 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)zbMATHGoogle 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)CrossRefzbMATHGoogle 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)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Felsner, S., Weil, H.: A theorem on higher Bruhat orders. Discrete Comput. Geom. 23(1), 121–127 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Hohlweg, C., Lange, C.E.M.C.: Realizations of the associahedron and cyclohedron. Discrete Comput. Geom. 37(4), 517–543 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press, Cambridge (1992)zbMATHGoogle Scholar
  14. 14.
    Jonsson, J.: Generalized triangulations and diagonal-free subsets of stack polyominoes. J. Comb. Theory, Ser. A 112(1), 117–142 (2005)CrossRefzbMATHGoogle Scholar
  15. 15.
    Knutson, A., Miller, E.: Subword complexes in Coxeter groups. Adv. Math. 184(1), 161–176 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Knutson, A., Miller, E.: Gröbner geometry of Schubert polynomials. Ann. Math. 161(3), 1245–1318 (2005)CrossRefzbMATHMathSciNetGoogle 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)zbMATHGoogle Scholar
  19. 19.
    Pilaud, V., Santos, F.: Multitriangulations as complexes of star polygons. Discrete Comput. Geom. 41(2), 284–317 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Pilaud, V., Pocchiola, M.: Multitriangulations, pseudotriangulations and primitive sorting networks. Discrete Comput. Geom. 48(1), 142–191 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Pilaud, V., Santos, F.: The brick polytope of a sorting network. Eur. J. Comb. 33(4), 632–662 (2012)CrossRefzbMATHMathSciNetGoogle 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)CrossRefzbMATHMathSciNetGoogle 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)CrossRefzbMATHMathSciNetGoogle 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)CrossRefzbMATHMathSciNetGoogle 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