A positive Monk formula in the S1‐equivariant cohomology of type A Peterson varieties

By Megumi Harada, Julianna Tymoczko. 2009

51 citations recorded by the source corpus.

Peterson varieties are a special class of Hessenberg varieties that have been extensively studied, for example, 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 An−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 HS1* (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 HS1* (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 HS1* (Y) and the ordinary cohomology ring H*(Y) of the type An−1 Peterson variety. Our methods are both directly from and inspired by those of the GKM (Goresky–Kottwitz–MacPherson) theory and classical Schubert calculus. We discuss several open questions and directions for future work.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/81c9ac398cac219218e3ffb18d241fa2f5d4a079

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In Discrete Mathematics and Combinatorics

1. Megumi Harada
2. M. Masuda
3. T. Horiguchi
4. Hiraku Abe
5. Julianna Tymoczko
6. T. Panov
7. D. Suh
8. Martha Precup

In Geometry and Topology

1. Megumi Harada
2. M. Masuda
3. T. Horiguchi
4. Hiraku Abe
5. Julianna Tymoczko
6. T. Panov
7. D. Suh
8. Martha Precup

Citations in the corpus

• The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A (2015, methodology): “To cite just two examples, the second author and Tymoczko gave an explicit set of generators for H(Petn) and prove a Schubert-calculus-type “Monk formula”, resulting in a presentation of H(Petn) via generators and relations in [22], and in a different direction, Brion and Carrell showed an isomorphi”
• Poset pinball, GKM-compatible subspaces, and Hessenberg varieties (2010, methodology): “There has been much interest in this circle of ideas, especially in relation to the study of Hessenberg varieties, since the authors first introduced them in [23].”