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Subword complexes, cluster complexes, and generalized multi-associahedra. (English) Zbl 1286.05180
Summary: 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.

MSC:
05E15 Combinatorial aspects of groups and algebras (MSC2010)
52B11 \(n\)-dimensional polytopes
20F55 Reflection and Coxeter groups (group-theoretic aspects)
52B45 Dissections and valuations (Hilbert’s third problem, etc.)
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