Geometry, Complexity, and Combinatorics of Permutation Polytopes

Each group G of permutation matrices gives rise to a permutation polytope P(G) = cony(G) c Re×d, and for any x ~ W, an orbit polytope P(G, x) = conv(G, x). A broad subclass is formed by the Young permutation polytopes, which correspond bijectively to partitions 2 = (21, ..., 2k)~-n of positive integers, and arise from the Young representations of the symmetric group. Young polytopes provide a framework allowing a unified study of many combinatorial optimization problems of different computational complexities. In particular, the much studied traveling salesman polytope is a certain Young orbit polytope, and many decision problems, such as simplical complex isomorphism, reduce to optimizing linear functionals over Young polytopes. First, the classical polytope of bistochastic matrices P(Sn) = P((n-1, 1)) is studied. Large stable sets in its 1-skeleton, induced by the Young representations, are exhibited, and it is shown that its stability number c~(n) is 2 o('/~°gn). Next, we study low dimensional skeletons of Young polytopes in general. Letting m be the largest integer for which P(2) is m-neighborly, under some restrictions on 2 it is shown that [_kZ/2J ~<m < ½(k + 1)!. Finally, we study the following semialgebraic geometric question, posed by D. Kozen: Is the combinatorial type of the polytope, and oriented matroid, of a generic orbit, unique? We show that, while a theorem of Rado implies a positive answer for the symmetric group, the general answer is negative, and the induced stratifications are nontrivial, and should be the subject of a future study.