Subword complexes, cluster complexes, and generalized multi-associahedra
TL;DR: It is shown that multi-cluster complexes of types A and B coincide with known simplicial complexes, namely with the simplicialcomplex of multi-triangulations and centrally symmetric multi-Triangulations, respectively.
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Abstract: 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.
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Brick polytopes of spherical subword complexes and generalized associahedra
Vincent Pilaud,Christian Stump +1 more
TL;DR: In this article, the authors generalize the brick polytope of V. Pilaud and F. Santos to spherical subword complexes for finite Coxeter groups, yielding vertex description of generalized associahedra, a Minkowski sum decomposition into Coxeter matroid polytopes, and a combinatorial description of the exchange matrix of any cluster in a finite type cluster algebra.
Three non-equivalent realizations of the associahedron
Cesar Ceballos,Günter M. Ziegler +1 more
TL;DR: In this paper, the authors classify the associahedra obtained by these constructions modulo linear equivalence of their normal fans and show, in particular, that the only realization that can be obtained with both methods is the Chapoton-Fomin-Zelevinsky (2002).
54
The brick polytope of a sorting network
TL;DR: In this article, the authors construct the "brick polytope" of a network, obtained as the convex hull of the pseudoline arrangement supported by the network, and characterize its vertices, describe its faces, and decompose it as a Minkowski sum of simpler polytopes.
46
Multi-triangulations as complexes of star polygons
Vincent Pilaud,Francisco Santos +1 more
TL;DR: In this article, a new way of looking at $k$-triangulations is presented, where certain star polygons naturally generalize triangles for the convex polygon.
43
Denominator vectors and compatibility degrees in cluster algebras of finite type
Cesar Ceballos,Vincent Pilaud +1 more
TL;DR: In this article, the authors present two simple descriptions of the denominator vectors of the cluster variables of a cluster algebra of finite type, with respect to any initial cluster seed, in terms of the compatibility degrees between almost positive roots defined by S. Fomin and A. Zelevinsky.
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