Document Type
Article
Publication Date
1-2023
Publication Title
Pure and Applied Math Quarterly
Abstract
Hessenberg varieties H(X, H) form a class of subvarieties of the flag variety G/B, parameterized by an operator X and certain subspaces H of the Lie algebra of G. We identify several families of Hessenberg varieties in type An−1 that are T -stable subvarieties of G/B, as well as families that are invariant under a subtorus K of T . In particular, these varieties are candidates for the use of equivariant methods to study their geometry. Indeed, we are able to show that some of these varieties are unions of Schubert varieties, while others cannot be such unions. Among the T -stable Hessenberg varieties, we identify several that are GKM spaces, meaning T acts with isolated fixed points and a finite number of one- dimensional orbits, though we also show that not all Hessenberg varieties with torus actions and finitely many fixed points are GKM. We conclude with a series of open questions about Hessenberg varieties, both in type An−1 and in general Lie type.
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Rights
Licensed to Smith College and distributed CC-BY 4.0 under the Smith College Faculty Open Access Policy.
Recommended Citation
Golding, Rebecca and Tymoczko, Julianna, "Which Hessenberg varieties are GKM?" (2023). Mathematics Sciences: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/mth_facpubs/185
Comments
Authors' submitted manuscript.