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arXiv · 2605.01332

Toric Schubert Varieties in Partial Flag Varieties

Mahir Bilen Can·Arpita Nayek·Pinakinath Saha
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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.

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Mahir Bilen Can, Arpita Nayek, Pinakinath Saha. 2026-05-02. Toric Schubert Varieties in Partial Flag Varieties. https://arxiv.org/abs/2605.01332

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