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 n × n . 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 { 2 t , ⌊ 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 ) 2 precisely when G is 2-transitive. We then extend the results of Pak [I. Pak, Four questions on Birkhoff polytope, Ann. Comb. 4 (1) (2000) 83–90] 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.