Symmetric Alcoved Polytopes

  • Annette Werner
  • Josephine Yu
Keywords: polytopes, root systems, tropical geometry


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.

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