Toroidal Schubert Varieties

TL;DR

This paper investigates the actions of Levi subgroups on Schubert varieties, characterizing toroidal and horospherical cases, and analyzing their singular loci, with specific results in type A and for minuscule varieties.

Contribution

It provides a classification of toroidal and horospherical partial flag varieties with Picard number 1 and establishes necessary conditions for toroidal Levi subgroup actions on Schubert varieties.

Findings

Horospherical actions are determined for partial flag varieties.

The singular locus of (co)minuscule Schubert varieties contains all $L_{max}$-stable subvarieties.

In type A, Billey-Postnikov decomposition influences toroidal Schubert varieties.

Abstract

Levi subgroup actions on Schubert varieties are studied. In the case of partial flag varieties, the horospherical actions are determined. This leads to a characterization of the toroidal and horospherical partial flag varieties with Picard number 1. In the more general case, we provide a set of necessary conditions for the action of a Levi subgroup on a Schubert variety to be toroidal. The singular locus of a (co)minuscule Schubert variety is shown to contain all the $L_{max}$-stable Schubert subvarieties, where $L_{max}$ is the standard Levi subgroup of the maximal parabolic which acts on the Schubert variety by left multiplication. In type A, the effect of the Billey-Postnikov decomposition on toroidal Schubert varieties is obtained.