Abstract

We develop the theory of poset pinball, a combinatorial game introduced by Harada-Tymoczko to study the equivariant cohomology ring of a GKM-compatible subspace X of a GKM space; Harada and Tymoczko also prove that, in certain circumstances, a successful outcome of Betti poset pinball yields a module basis for the equivariant cohomology ring of X. First we define the dimension pair algorithm, which yields a successful outcome of Betti poset pinball for any type A regular nilpotent Hessenberg and any type A nilpotent Springer variety, considered as GKM-compatible subspaces of the flag variety. The algorithm is motivated by a correspondence between Hessenberg affine cells and certain Schubert polynomials which we learned from Insko. Second, in a special case of regular nilpotent Hessenberg varieties, we prove that our pinball outcome is poset-upper-triangular, and hence the corresponding classes form a HS1*(pt)-module basis for the S1-equivariant cohomology ring of the Hessenberg variety.

Highlights

  • The purpose of this paper is to further develop the theory of poset pinball, a combinatorial game introduced in 1 for the purpose of computing in equivariant cohomology rings all cohomology rings in this note are with C coefficients, in certain cases of type A nilpotent Hessenberg varieties

  • We develop the theory of poset pinball, a combinatorial game introduced by Harada-Tymoczko to study the equivariant cohomology ring of a GKM-compatible subspace X of a GKM space; Harada and Tymoczko prove that, in certain circumstances, a successful outcome of Betti poset pinball yields a module basis for the equivariant cohomology ring of X

  • In a special case of regular nilpotent Hessenberg varieties, we prove that our pinball outcome is poset-uppertriangular, and the corresponding classes form a HS∗1 pt -module basis for the S1-equivariant cohomology ring of the Hessenberg variety

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Summary

Introduction

The purpose of this paper is to further develop the theory of poset pinball, a combinatorial game introduced in 1 for the purpose of computing in equivariant cohomology rings all cohomology rings in this note are with C coefficients , in certain cases of type A nilpotent Hessenberg varieties. The results of 13 accomplish precisely this goal—that is, of constructing a module basis via poset pinball techniques—in the special case of the Peterson variety, which is the regular nilpotent Hessenberg variety with Hessenberg function h defined by h i i 1 for 1 ≤ i ≤ n − 1 and h n n Exploiting this explicit module basis, in 13, Theorem 6.12 the second author and Tymoczko give a manifestly positive Monk formula for the product of a degree-2 Peterson Schubert class with an arbitrary Peterson Schubert class, expressed as a HS∗1 pt linear combination of Peterson Schubert classes.

Background
Bijection between Fixed Points and Permissible Fillings
The Dimension Pair Algorithm for Nilpotent Hessenberg Varieties
Fixed Points and Associated Subsets for the 334-Type Hessenberg Variety
Reduced Word Decompositions for 334-Type Fixed Points and Rolldowns
Bruhat Order Relations
Open Questions
Full Text
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