Vertices of FFLV polytopes

TL;DR

This paper investigates the vertices of FFLV polytopes associated with irreducible representations of Lie algebras, providing new classifications and enumerations of these vertices for type A and symplectic cases.

Contribution

It characterizes various sets of vertices of FFLV polytopes, including permutation and simple vertices, and establishes their properties and counts, extending results to symplectic algebras.

Findings

Vertices are local with respect to type A Dynkin diagram.

Number of simple vertices equals the large Schr"oder number.

Analogous vertex results are derived for symplectic algebras.

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 Schr\"oder number. Finally, we derive analogous results for symplectic algebras.