Spherical Schubert varieties and pattern avoidance
Christian Gaetz

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 is called -spherical for the action of the complex reductive group if it contains a dense orbit of some Borel subgroup of . 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 with respect to the largest possible Levi subgroup is characterized by this same pattern avoidance condition.
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