# zbMATH — the first resource for mathematics

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)
##### Keywords:
generalized associahedra; root systems
Full Text: