Poset Pinball, the Dimension Pair Algorithm, and Type A Regular 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 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 $H_{S^1}^{*}$ (pt)-module basis for the S 1 -equivariant cohomology ring of the Hessenberg variety.