Abstract
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.
Highlights
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
We show that for any root system other than F4, 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
In this paper we investigate alcoved polytopes which are symmetric under the action of the Weyl group
Summary
In this paper we investigate alcoved polytopes which are symmetric under the action of the Weyl group. For a root system Φ, an alcoved polytope of type Φ is a polytope defined by inequalities of the form a, x c where a ∈ Φ and c ∈ Z They are unions of (faces of) alcoves in the affine Coxeter arrangement associated to Φ. For type A root systems, alcoved polytopes are precisely the tropical polytopes that are convex in the usual sense. They are named polytropes by Joswig and Kulas [Jo-Ku]. Our main result states that every alcoved polytope for an irreducible and reduced root system Φ not of type F4 that is symmetric under the action of the Weyl group can be generated by h vertices, where h is the Coxeter number of Φ.
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