Faces of Birkhoff Polytopes

The Birkhoff polytope $B_n$ is the convex hull of all $(n\times n)$ 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 $\mathcal L$ of faces of a Birkhoff polytope. The Birkhoff dimension $\mathrm{bd}(\mathcal L)$ of $\mathcal L$ is the smallest $n$ such that $B_n$ has a face with combinatorial type $\mathcal L$.By a result of Billera and Sarangarajan, a combinatorial type $\mathcal L$ of a $d$-dimensional face appears in some $\mathcal B_k$ for $k\le 2d$, so $\mathrm{bd}(\mathcal L)\le 2d$. We will characterize those types with $\mathrm{bd}(\mathcal L)\ge 2d-3$, and we prove that any type with $\mathrm{bd}(\mathcal L)\ge d$ is either a product or a wedge over some lower dimensional face. Further, we computationally classify all $d$-dimensional combinatorial types for $2\le d\le 8$.