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Symmetric alcoved polytopes. (English) Zbl 1302.52014
Summary: Generalized alcoved polytopes are polytopes whose facet normals are roots in a given root system. We call a set of points in an alcoved polytope a generating set if there does not exist a strictly smaller alcoved polytope containing it. The type $$A$$ alcoved polytopes are precisely the tropical polytopes that are also convex in the usual sense. In this case the tropical generators form a generating set. We show that for any root system other than $$F_4$$, every alcoved polytope invariant under the natural Weyl group action has a generating set of cardinality equal to the Coxeter number of the root system.

##### MSC:
 52B15 Symmetry properties of polytopes 14T05 Tropical geometry (MSC2010) 17B22 Root systems
##### Keywords:
polytopes; root systems; tropical geometry
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