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.
Categories: math.CO math.GR math.RT