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

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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 \(\varDelta _{11,3}\).

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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.

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Authors and Affiliations

  1. Fields Institute, York University, Toronto, ON, Canada

    Nantel Bergeron & Cesar Ceballos

  2. Institut für Mathematik, Freie Universität Berlin, Arnimallee 2, 14195, Berlin, Germany

    Jean-Philippe Labbé

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  1. Nantel Bergeron
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  2. Cesar Ceballos
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  3. Jean-Philippe Labbé
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Correspondence to Jean-Philippe Labbé.

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Appendix: Counting Matrices and Their Gale Dual Matrices: General Formulas

Appendix: Counting Matrices and Their Gale Dual Matrices: General Formulas

In this appendix, we present general formulas for the counting matrices \(D_{c,m}\) of type \(A_n\) with \(n\le 3\). We also present formulas for their Gale dual matrices, which are used to exhibit explicit coordinates for the complete simplicial fans realizing the subword complexes and multi-associahedra of type \(A_3\) in Corollaries 1 and 2.

1.1 Counting Matrices

The matrices \(D_{c,m}\) of type \(A_n\) with \(n\le 3\) are given below:

Type \(A_1\), \(c=(s_1)\):

$$\begin{aligned} D_{c,m} = \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 1&1&\ldots&1&1&1 \end{array} \right) _{1\times m}. \end{aligned}$$

Type \(A_2\), \(c=(s_1,s_2)\), with rows {\(\alpha _1\), \(\alpha _1+\alpha _2\), \(\alpha _2\)} in this order:

$$\begin{aligned} D_{c,m} =\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 1&{} 0 &{} 1 &{} 0 &{}\ldots &{} 1 &{} 0 \\ m &{} 1 &{} m-1 &{} 2 &{}\ldots &{} 1 &{} m \\ 0 &{} 1 &{} 0 &{} 1 &{}\ldots &{} 0 &{} 1 \end{array} \right) _{3\times 2m}. \end{aligned}$$

Type \(A_3\), \(c=(s_2,s_1,s_3)\), with rows {\(\alpha _2\), \(\alpha _1+\alpha _2\), \(\alpha _2+\alpha _3\), \(\alpha _1+\alpha _2+\alpha _3\), \(\alpha _3\), \(\alpha _1\)} in this order:

Type \(A_3\), \(c=(s_1,s_2,s_3)\), with rows {\(\alpha _1\), \(\alpha _1+\alpha _2\), \(\alpha _1+\alpha _2+\alpha _3\), \(\alpha _2\), \(\alpha _2+\alpha _3\), \(\alpha _3\)} in this order:

1.2 Gale Dual Matrices

Below are explicit choices for Gale duals matrices \(M_{c,m}\) of \(D_{c,m}\).

Type \(A_1\), \(c=(s_1)\):

Type \(A_2\), \(c=(s_1,s_2)\):

Type \(A_3\), \(c=(s_2,s_1,s_3)\): Matrix \(M_{213,m}\) in Corollary 1.

Type \(A_3\), \(c=(s_1,s_2,s_3)\): Matrix \(M_{123,m}\) in Corollary 2.

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Bergeron, N., Ceballos, C. & Labbé, JP. Fan Realizations of Type \(A\) Subword Complexes and Multi-associahedra of Rank 3. Discrete Comput Geom 54, 195–231 (2015). https://doi.org/10.1007/s00454-015-9691-0

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