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
Summary
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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have