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Brick polytopes of spherical subword complexes and generalized associahedra. (English) Zbl 1405.05196
Summary: We generalize the brick polytope of the first author and F. Santos [Discrete Comput. Geom. 41, No. 2, 284–317 (2009; Zbl 1177.52007); Eur. J. Comb. 33, No. 4, 632–662 (2012; Zbl 1239.52026)] to spherical subword complexes for finite Coxeter groups. This construction provides polytopal realizations for a certain class of subword complexes containing all cluster complexes of finite types. For the latter, the brick polytopes turn out to coincide with the known realizations of generalized associahedra, thus opening new perspectives on these constructions. This new approach yields in particular the 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.

MSC:
05E45 Combinatorial aspects of simplicial complexes
05E15 Combinatorial aspects of groups and algebras (MSC2010)
05E30 Association schemes, strongly regular graphs
20F55 Reflection and Coxeter groups (group-theoretic aspects)
52B12 Special polytopes (linear programming, centrally symmetric, etc.)
13F60 Cluster algebras
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