Faces of birkhoff polytopes

Abstract

The Birkhoff polytope Bn is the convex hull of all (n x n) permutation matrices, i.e., matrices where precisely one entry in each row and column is one, and zeros at all other places. This is a widely studied polytope with various applications throughout mathematics. In this paper we study combinatorial types C of faces of a Birkhoff polytope. The Birkhoff dimension bd(£) of C is the smallest n such that Bn has a face with combinatorial type C. By a result of Billera and Sarangarajan, a combinatorial type C of a d-dimensio-nal face appears in some Bk for k≤ 2d, so bd(£) ≤ 2d. We will characterize those types with bd(£) ≥ 2d — 3, and we prove that any type with bd(£) ≥ d is either a product or a wedge over some lower dimensional face. Further, we computationally classify all d-dimensional combinatorial types for 2 ≤ d ≤ 8.