University of Toronto's Symplectic Geometry Seminar

October 26, 2009, 2:10 pm
Bahen 6183

Megumi Harada

McMaster University

A generalized Schubert calculus for type A Peterson varieties


Hessenberg varieties are certain subvarieties of flag varieties G/B which appear in a wide array of mathematical areas, including geometric representation theory, numerical analysis, algebraic geometry, and combinatorics. Peterson varieties are a special case, which have been extensively studied by e.g. Peterson, Kostant, and Rietsch in connection with the quantum cohomology of the flag variety. In this talk I will explain recent joint work with Julianna Tymoczko, in which we develop a `generalized Schubert calculus' in the S^1-equivariant cohomology of Peterson varieties. More specifically, we identify a computationally convenient subset of the well-known equivariant Schubert classes \sigma_w in the T-equivariant cohomology of G/B which then projects to a module basis of the S^1-equivariant cohomology of the Peterson variety. Moreover, using the restriction data at the S^1-fixed points and techniques inspired from GKM (= Goresky-Kottwitz-MacPherson) theory, we derive explicit, combinatorial (and hence manifestly positive and manifestly integral) formulas for sufficiently many structure constants for the S^1-equivariant cohomology ring of Peterson varieties to completely determine the ring. In keeping with Schubert calculus tradition, we call these ``Chevalley-Monk formulae". Our techniques are clearly related to, but also clearly different from, those recently developed by Goldin and Tolman in the context of equivariant symplectic geometry (for example, Goldin-Tolman always assume their spaces are manifolds, while Peterson varieties are in general singular). As far as we are aware, this is the first example of a complete and combinatorial `Schubert calculus' for any variety which is not a Kac-Moody homogeneous space G/P. Indeed, we view these results as the first steps in the development of an extensive theory of generalized Schubert calculus beyond the realm of the G/P; for instance, in future work, we expect to deal with the case of general regular nilpotent Hessenberg varieties, as well as certain subspaces of GKM spaces. The talk will be aimed at graduate students. In particular, at the beginning I will give an impressionistic sketch of Schubert calculus in general. I will then introduce the main characters, which are the flag varieties and the Peterson varieties (as well as the torus actions on them), and give an idea of the GKM and Schubert calculus techniques which enter into the proofs of the claims above. Time permitting, I will conclude with a sampling of open questions.