Faces of the signed Birkhoff polytopes

We study faces of the signed Birkhoff polytopes, denoted by $\Omega_n^{\pm}$. We describe its nonempty faces, $1$-dimensional faces, $2$-dimensional faces, and facets. Moreover, we study the diameter and Hamiltonian connectivity of the graph of $\Omega_n^{\pm}$. In the end, we show that the reduced Gröbner basis of the toric ideal of the signed Birkhoff polytope $\Omega_n^{\pm}$ with respect to the graded reverse lexicographic order induced by rank orders has square-free initial monomials of degree $\leq n$.