  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.
