Document Type
Article
Publication Date
5-1-2010
Publication Title
Journal of Algebra
Abstract
Regular semisimple Hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory, numerical analysis, representation theory, algebraic geometry, and combinatorics. We give a " Giambelli formula" expressing the classes of regular semisimple Hessenberg varieties in terms of Chern classes. In fact, we show that the cohomology class of each regular semisimple Hessenberg variety is the specialization of a certain double Schubert polynomial, giving a natural geometric interpretation to such specializations. We also decompose such classes in terms of the Schubert basis for the cohomology ring of the flag variety. The coefficients obtained are nonnegative, and we give closed combinatorial formulas for the coefficients in many cases. We introduce a closely related family of schemes called regular nilpotent Hessenberg schemes, and use our results to determine when such schemes are reduced.
Keywords
Degeneracy locus, Hessenberg variety, Schubert polynomial, Schubert variety
Volume
323
Issue
10
First Page
2605
Last Page
2623
DOI
10.1016/j.jalgebra.2010.03.001
ISSN
00218693
Recommended Citation
Anderson, Dave and Tymoczko, Julianna, "Schubert Polynomials and Classes of Hessenberg Varieties" (2010). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/mth_facpubs/106
Comments
Peer reviewed accepted manuscript.