Poset Pinball, the Dimension Pair Algorithm, and TypeARegular Nilpotent Hessenberg Varieties

摘要

We develop the theory ofposet pinball, a combinatorial game introduced by Harada-Tymoczko to study the equivariant cohomology ring of a GKM-compatible subspaceXof a GKM space; Harada and Tymoczko also prove that, in certain circumstances, asuccessful outcome of Betti poset pinballyields a module basis for the equivariant cohomology ring ofX. First we define thedimension pair algorithm, which yields a successful outcome of Betti poset pinball for any typeAregular nilpotent Hessenberg and any typeAnilpotent 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 isposet-upper-triangular, and hence the corresponding classes form aHS1*(pt)-module basis for theS1-equivariant cohomology ring of the Hessenberg variety.