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Polytopal realizations of generalized associahedra. (English) Zbl 1018.52007
Let \(\Phi\) be a rank \(n\) finite root system with the set of simple roots \(\Pi=\{\alpha_i\;|\;i\in I\}\) and the set of positive roots \(\Phi_{>0}\). Let \(\Phi_{\geq -1}=\Phi_{>0}\cup (-\Pi)\). The simplicial complex \(\Delta(\Phi)\) (generalized associahedron) has \(\Phi_{\geq -1}\) as set of vertices; its simplices are the subsets of mutually compatible elements of \(\Phi_{\geq -1}\). The maximal simplices of \(\Delta(\Phi)\) are called clusters.
The main result of this paper is Theorem 1.4. The simplicial fan \(\Delta(\Phi)\) is the normal fan of a simple \(n\)-dimensional convex polytope.

MSC:
52B11 \(n\)-dimensional polytopes
51E12 Generalized quadrangles and generalized polygons in finite geometry
20F55 Reflection and Coxeter groups (group-theoretic aspects)
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
05E15 Combinatorial aspects of groups and algebras (MSC2010)
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