Faces of Birkhoff Polytopes

TL;DR

This paper investigates the combinatorial types of faces of Birkhoff polytopes, characterizes those with high Birkhoff dimension, and computationally classifies types for dimensions 2 to 8.

Contribution

It characterizes face types with high Birkhoff dimension and provides a computational classification for dimensions 2 to 8.

Findings

Characterization of face types with Birkhoff dimension at least 2d-3.

Proof that high-dimensional types are products or wedges of lower-dimensional faces.

Complete classification of combinatorial face types for dimensions 2 through 8.

Abstract

The Birkhoff polytope B(n) 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 L of faces of a Birkhoff polytope. The Birkhoff dimension bd(L) of L is the smallest n such that B(n) 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 B(k) for k less or equal to 2d, so bd(L) is at most d. We will characterize those types whose Birkhoff dimension is at least 2d-3, and we prove that any type whose Birkhoff dimension is at least d is either a product or a wedge over some lower dimensional face. Further, we computationally classify all d-dimensional combinatorial types for d between 2 and 8.