Toric Schubert Varieties in Partial Flag Varieties

TL;DR

This paper characterizes toric Schubert varieties within partial flag varieties, providing explicit fan descriptions, smoothness criteria, and lattice structures, with applications to spherical and horospherical varieties.

Contribution

It offers a combinatorial framework for understanding the geometry and smoothness of toric Schubert varieties in partial flag varieties, extending previous work with explicit fan descriptions and lattice analysis.

Findings

Explicit fan descriptions for toric Schubert varieties.

Necessary and sufficient conditions for smoothness based on Cartan integers.

Interval $[e,w]_{W^P}$ forms a supersolvable join-distributive lattice.

Abstract

In this article, we investigate the toric Schubert varieties in partial flag varieties $G/P$ for a connected semisimple algebraic group $G$. Using Deodhar's decomposition of Richardson varieties and the work of Pasquier, we give an explicit description of the fan of a toric Schubert variety, leading to a combinatorial model for its cones. As an application, we obtain necessary and sufficient conditions for smoothness of toric Schubert varieties in terms of the Cartan integers associated to a reduced expression. Furthermore, we prove that for a Coxeter-type element $w \in W^P$, the interval $[e,w]_{W^P}$ is a supersolvable join-distributive lattice. Finally, we apply these results to the study of spherical and horospherical Schubert varieties, providing a combinatorial method for checking the smoothness via the associated toric Schubert varieties.