Linear Conditions Imposed on Flag Varieties

Authors

Julianna S. Tymoczko, University of Michigan, Ann ArborFollow

Document Type

Article

Publication Date

12-1-2006

Publication Title

American Journal of Mathematics

Abstract

We study subvarieties of the flag variety called Hessenberg varieties, defined by certain linear conditions. These subvarieties arise naturally in applications including geometric representation theory, number theory, and numerical analysis. We describe completely the homology of Hessenberg varieties over GLn(ℂ) and show that they have no odd-dimensional homology. We provide an explicit geometric construction which partitions each Hessenberg variety into pieces homeomorphic to affine space. We characterize these affine pieces by fillings of Young tableaux and show that the dimension of the affine piece can be computed by combinatorial rules generalizing the Eulerian numbers. We give an equivalent formulation of this result in terms of roots. We conclude with a section on open questions.

Volume

128

Issue

6

First Page

1587

Last Page

1604

DOI

10.1353/ajm.2006.0050

ISSN

00029327

Comments

Peer reviewed accepted manuscript.

Recommended Citation

Tymoczko, Julianna S., "Linear Conditions Imposed on Flag Varieties" (2006). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/mth_facpubs/110