Permutation polytopes and indecomposable elements in permutation groups

Each group G of n × n permutation matrices has a corresponding permutation polytope, P (G) := conv(G) ⊂ R. We relate the structure of P (G) to the transitivity of G. In particular, we show that if G has t nontrivial orbits, then min{2t, ⌊n/2⌋} is a sharp upper bound on the diameter of the graph of P (G). We also show that P (G) achieves its maximal dimension of (n − 1) precisely when G is 2-transitive. We then extend the results of Pak [22] on mixing times for a random walk on P (G). Our work depends on a new result for permutation groups involving writing permutations as products of indecomposable permutations.