Poset pinball, GKM-compatible subspaces, and Hessenberg varieties

By Megumi Harada, Julianna Tymoczko. 2010

48 citations recorded by the source corpus.

This paper has three main goals. First, we set up a general framework to address the problem of constructing module bases for the equivariant cohomology of certain subspaces of GKM spaces. To this end we introduce the notion of a GKM-compatible subspace of an ambient GKM space. We also discuss poset-upper-triangularity, a key combinatorial notion in both GKM theory and more generally in localization theory in equivariant cohomology. With a view toward other applications, we present parts of our setup in a general algebraic and combinatorial framework. Second, motivated by our central problem of building module bases, we introduce a combinatorial game which we dub poset pinball and illustrate with several examples. Finally, as first applications, we apply the perspective of GKM-compatible subspaces and poset pinball to construct explicit and computationally convenient module bases for the $S^1$-equivariant cohomology of all Peterson varieties of classical Lie type, and subregular Springer varieties of Lie type $A$. In addition, in the Springer case we use our module basis to lift the classical Springer representation on the ordinary cohomology of subregular Springer varieties to $S^1$-equivariant cohomology in Lie type $A$.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/de0fc6ddadf4cee0a0b167ef26c42491196bb2a3

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

1. M. Masuda
2. T. Horiguchi
3. Hiraku Abe
4. Megumi Harada
5. T. Panov
6. D. Suh
7. Suyoung Choi
8. V. Buchstaber

In Geometry and Topology

1. M. Masuda
2. T. Horiguchi
3. Hiraku Abe
4. Megumi Harada
5. T. Panov
6. D. Suh
7. Suyoung Choi

Citations in the corpus

• The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A (2015, background): “ and Tymoczko have provided an explicit set of generators for H∗ S(Petn) and also prove a Schubert-calculus-type “Monk formula”, resulting in a presentation of H∗ S(Petn) via generators and relations [25]. In a different direction, Brion and Carrell showed an isomorphism between the equivariant cohom”