One-skeleton posets of Bruhat interval polytopes

TL;DR

This paper studies the 1-skeleton posets of Bruhat interval polytopes, proving they form lattices, classifying when these polytopes are simple, and determining conditions for smooth torus orbit closures in Schubert varieties.

Contribution

It establishes that the 1-skeleton posets are lattices and classifies the simplicity of Bruhat interval polytopes, resolving several open problems and conjectures.

Findings

1-skeleton posets of Bruhat interval polytopes are lattices

Classified when these polytopes are simple

Identified conditions for smooth torus orbit closures in Schubert varieties

Abstract

Introduced by Kodama and Williams, Bruhat interval polytopes 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 polytopes 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.