One-skeleton posets of Bruhat interval polytopes

Introduced by Kodama and Williams, Bruhat interval poly -topes are generalized permutohedra closely connected to the study of torus orbit closures and total positivity in Schubert varieties. We show that the 1-skeleton posets of these poly -topes are lattices and classify when the polytopes are simple, thereby resolving open problems and conjectures of Fraser, of Lee-Masuda, and of Lee-Masuda-Park. In particular, we classify when generic torus orbit closures in Schubert varieties are smooth. & COPY; 2023 Elsevier Inc. All rights reserved.