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Vertices of FFLV polytopes. (English) Zbl 1370.05218
Authors’ abstract: FFLV polytopes describe monomial bases in irreducible representations of $$\mathfrak {sl}_n$$ and $$\mathfrak {sp}_{2n}$$. We study various sets of vertices of FFLV polytopes. First, we consider the special linear case. We prove the locality of the set of vertices with respect to the type $$A$$ Dynkin diagram. Then we describe all the permutation vertices and after that we describe all the simple vertices and prove that their number is equal to the large Schröder number. Finally, we derive analogous results for symplectic Lie algebras.

##### MSC:
 05E10 Combinatorial aspects of representation theory 52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) 52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) 17B56 Cohomology of Lie (super)algebras
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