Discrete & Computational Geometry

, Volume 54, Issue 1, pp 195–231

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

  • Nantel Bergeron
  • Cesar Ceballos
  • Jean-Philippe Labbé

DOI: 10.1007/s00454-015-9691-0

Cite this article as:
Bergeron, N., Ceballos, C. & Labbé, JP. Discrete Comput Geom (2015) 54: 195. doi:10.1007/s00454-015-9691-0


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 

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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