Classification of Levi-spherical Schubert varieties

TL;DR

This paper provides a combinatorial classification of Levi-spherical Schubert varieties in the complete flag manifold, confirming a conjecture and introducing a new Coxeter element formulation using key polynomial theory.

Contribution

It establishes a complete combinatorial classification of Levi-spherical Schubert varieties, confirming a prior conjecture and linking it to Coxeter elements and Demazure module characters.

Findings

Classification confirms the conjecture about Levi-spherical Schubert varieties.

New formulation in terms of standard Coxeter elements.

Uses key polynomial theory to prove the classification.

Abstract

A Schubert variety in the complete flag manifold $GL_n/B$ is Levi-spherical if the action of a Borel subgroup in a Levi subgroup of a standard parabolic has a dense orbit. We give a combinatorial classification of these Schubert varieties. This establishes a conjecture of the latter two authors, and a new formulation in terms of standard Coxeter elements. Our proof uses the theory of key polynomials (type A Demazure module characters).