Geometry, Complexity, and Combinatorics of Permutation Polytopes

Abstract Each group G of permutation matrices gives rise to a permutation polytope P(G) = conv( G ) ⊂ R d × d , and for any χϵ R d , an orbit polytope P ( G , χ ) = conv( G · χ ). A broad subclass is formed by the Young permutation polytopes , which correspond bijectively to partitions λ = ( λ 1 … λ k ) ⊢ 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 ( S n ) = 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 α ( n ) is 2 ω ( n logn) . Next, we study low dimensional skeletons of Young polytopes in general. Letting m be the largest integer for which P ( λ ) is m -neighborly, under some restrictions on λ it is shown that ⌊ k 2 2 ⌋ ⩽ m 1 2 (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.