Faces of Birkhoff Polytopes

The Birkhoff polytope Bn is the convex hull of all (n×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 L of faces of a Birkhoff polytope. The Birkhoff dimension bd(L) of L is the smallest n such that Bn has a face with combinatorial type L. By a result of Billera and Sarangarajan, a combinatorial type L of a d-dimensional face appears in some Bk for k 6 2d, so bd(L) 6 2d. We will characterize those types with bd(L) > 2d− 3, and we prove that any type with bd(L) > d is either a product or a wedge over some lower dimensional face. Further, we computationally classify all d-dimensional combinatorial types for 2 6 d 6 8.