Abstract
A Schubert variety in the complete flag manifold \(GL_n/B\) is Levi-spherical if the action of a Borel subgroup in a Levi subgroup of a standard parabolic has an open dense orbit. We give a combinatorial classification of these Schubert varieties. This establishes a conjecture of the latter two authors, and a new formulation in terms of standard Coxeter elements. Our proof uses and contributes to the theory of key polynomials (type A Demazure module characters).
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Notes
As mentioned in the introduction, in later work [16, Section 4] such a counterexample was indeed verified using Demazure character computations.
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Acknowledgements
We thank David Brewster, Jiasheng Hu, and Husnain Raza for writing useful computer code (in the NSF RTG funded ICLUE program). We also thank David Anderson, Mahir Can, Alexander Diaz-Lopez, Christian Gaetz, Megumi Harada, Bogdan Ion, Syu Kato, and Allen Knutson for stimulating conversations during the preparation of this work. We are grateful to the anonymous referees; their comments and suggestions led to improvements of this paper. We used SageMath, as well as the Maple packages ACE and Coxeter/Weyl in our investigations. This work was partially completed during (virtual) residence at ICERM’s Spring 2021 semester “Combinatorial Algebraic Geometry”; we thank the organizers providing an hospitable environment. AY was partially supported by a Simons Collaboration Grant, and an NSF RTG grant. RH was partially supported by an AMS-Simons Travel Grant.
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