Spherical Schubert varieties and pattern avoidance

TL;DR

This paper proves that spherical permutations and certain spherical Schubert varieties can be characterized by permutation pattern avoidance, resolving a conjecture and linking geometric properties to combinatorial patterns.

Contribution

It establishes a pattern avoidance criterion for spherical permutations and Schubert varieties, confirming a conjecture and connecting geometry with combinatorics.

Findings

Spherical permutations are characterized by pattern avoidance.

Spherical Schubert varieties' sphericality is determined by pattern avoidance.

The conjecture of Hodges--Yong is resolved through this characterization.

Abstract

A normal variety $X$ is called $H$-spherical for the action of the complex reductive group $H$ if it contains a dense orbit of some Borel subgroup of $H$. We resolve a conjecture of Hodges--Yong by showing that their spherical permutations are characterized by permutation pattern avoidance. Together with results of Gao--Hodges--Yong this implies that the sphericality of a Schubert variety $X_w$ with respect to the largest possible Levi subgroup is characterized by this same pattern avoidance condition.