Toric matrix Schubert varieties and their polytopes

TL;DR

This paper investigates when matrix Schubert varieties are toric, characterizes their associated polytopes, and constructs triangulations that realize subword complexes geometrically.

Contribution

It provides a characterization of toric matrix Schubert varieties and constructs their polytope triangulations linked to subword complexes.

Findings

Characterization of when $Y_$ is toric.

Construction of regular triangulations of the associated polytopes.

Connection of these triangulations to subword complexes.

Abstract

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.