A property of the Birkhoff polytope

TL;DR

This paper investigates the symmetry properties of the Birkhoff polytope and characterizes the uniqueness of the permutation matrix group in generating a polytope with the same face lattice.

Contribution

It computes the combinatorial symmetry group of the Birkhoff polytope and shows the uniqueness of the permutation matrix group among finite matrix groups with similar face lattice properties.

Findings

The symmetry group of the Birkhoff polytope is explicitly determined.

Permutation matrices are essentially the only finite matrix group producing a polytope with the same face lattice.

The result highlights a unique property of the Birkhoff polytope's structure.

Abstract

The Birkhoff polytope $B_n$ is the convex hull of all $n\times n$ permutation matrices in $\mathbb{R}^{n\times n}$. We compute the combinatorial symmetry group of the Birkhoff polytope. A representation polytope is the convex hull of some finite matrix group $G\leq \operatorname{GL}(d,\mathbb{R})$. We show that the group of permutation matrices is essentially the only finite matrix group which yields a representation polytope with the same face lattice as the Birkhoff polytope.