Toric matrix Schubert varieties and their polytopes

Given a matrix Schubert variety $\overline{X_\pi}$, it can be written as $\overline{X_\pi}=Y_\pi\times \mathbb{C}^q$ (where $q$ is maximal possible). We characterize when $Y_{\pi}$ is toric (with respect to a $(\mathbb{C}^*)^{2n-1}$-action) and study the associated polytope $\Phi(\mathbb{P}(Y_\pi))$ of its projectivization. We construct regular triangulations of $\Phi(\mathbb{P}(Y_\pi))$ which we show are geometric realizations of a family of subword complexes. Subword complexes were introduced by Knutson and Miller in 2004, who also showed that they are homeomorphic to balls or spheres and raised the question of their polytopal realizations.