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 



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.


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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

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