A Positive Monk Formula in the S¹-Equivariant Cohomology of Type A Peterson Varieties

Authors

Megumi Harada, McMaster University
Julianna Tymoczko, Smith CollegeFollow

Document Type

Article

Publication Date

6-30-2011

Publication Title

Proceedings of the London Mathematical Society

Abstract

Peterson varieties are a special class of Hessenberg varieties that have been extensively studied e.g. by Peterson, Kostant, and Rietsch, in connection with the quantum cohomology of the flag variety. In this manuscript, we develop a generalized Schubert calculus, and in particular a positive Chevalley-Monk formula, for the ordinary and Borel-equivariant cohomology of the Peterson variety Y in type A_n−1, with respect to a natural S¹ -action arising from the standard action of the maximal torus on flag varieties. As far as we know, this is the first example of positive Schubert calculus beyond the realm of Kac-Moody flag varieties G/P.

Our main results are as follows. First, we identify a computationally convenient basis of H^∗_S1(Y ), which we call the basis of Peterson Schubert classes. Second, we derive a manifestly positive, integral Chevalley-Monk formula for the product of a cohomology-degree-2 Peterson Schubert class with an arbitrary Peterson Schubert class. Both H^∗_S1(Y ) and H ^∗ (Y ) are generated in degree 2. Finally, by using our Chevalley-Monk formula we give explicit descriptions (via generators and relations) of both the S¹ -equivariant cohomology ring H^∗_S1(Y ) and the ordinary cohomology ring H ^∗ (Y ) of the type A_{n−1 Peterson variety. Our methods are both directly from and inspired by those of GKM (Goresky-Kottwitz-MacPherson) theory and classical Schubert calculus. We discuss several open questions and directions for future work.}

DOI

dx.doi.org/10.1112/plms/pdq038

ISSN

1460-244X

Comments

Peer reviewed accepted manuscript. Language included at the request of the publisher: this version differs from the final published version.

Recommended Citation

Harada, Megumi and Tymoczko, Julianna, "A Positive Monk Formula in the S¹-Equivariant Cohomology of Type A Peterson Varieties" (2011). Mathematics Sciences: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/mth_facpubs/13