Geodesic
Subword complexes, cluster complexes, and generalized multi-associahedra
Journal of Algebraic Combinatorics, Tome 39 (2014) no. 1, pp. 17-51.

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In this paper, we use subword complexes to provide a uniform approach to finite-type cluster complexes and multi-associahedra. We introduce, for any finite Coxeter group and any nonnegative integer $k$, a spherical subword complex called multi-cluster complex. For $k=1$, we show that this subword complex is isomorphic to the cluster complex of the given type. We show that multi-cluster complexes of types $A$ and $B$ coincide with known simplicial complexes, namely with the simplicial complexes of multi-triangulations and centrally symmetric multi-triangulations, respectively. Furthermore, we show that the multi-cluster complex is universal in the sense that every spherical subword complex can be realized as a link of a face of the multi-cluster complex.
Classification : 05E15, 52B11, 20F55, 52B45
Keywords: subword complex, cluster complex, generalized associahedron, multi-triangulation, Auslander-Reiten quiver, Coxeter-Catalan combinatorics 
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     title = {Subword complexes, cluster complexes, and generalized multi-associahedra},
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Ceballos, Cesar; Labbé, Jean-Philippe; Stump, Christian. Subword complexes, cluster complexes, and generalized multi-associahedra. Journal of Algebraic Combinatorics, Tome 39 (2014) no. 1, pp. 17-51. http://geodesic.mathdoc.fr/item/JAC_2014__39_1_a8/