Hessenberg varieties associated to ad-nilpotent ideals

Abstract

We consider Hessenberg varieties in the flag variety of $G{L}_n(\mathbb{C})$ with the property that the corresponding Hessenberg function defines an ad-nilpotent ideal. Each such Hessenberg variety is contained in a Springer fiber. We extend a theorem of Tymoczko to this setting, showing that these varieties have an affine paving obtained by intersecting with Schubert cells. Our method of proof constructs an affine paving for each Springer fiber that restricts to an affine paving of the Hessenberg variety. We use the combinatorial properties of this paving to prove that Hessenberg varieties of this kind are connected.

Keywords:

• Affine paving
• flag variety
• Hessenberg variety
• Springer fiber
• Young diagram

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 14M15
• 14L35
• 05A05

Acknowledgements

The authors are thankful to the anonymous referee for their feedback, which improved the exposition of Section 5.

Additional information

Funding

The second author is supported in part by NSF DMS 1954001.