Poset Pinball, the Dimension Pair Algorithm, and Type A Regular Nilpotent Hessenberg Varieties. (10th June 2012)

• Record Type: Journal Article
• Authors: Bayegan, Darius; Harada, Megumi
• Other Names: Jeffrey L. C. Academic Editor.; Morozov A. Academic Editor.; Saidi E. H. Academic Editor.
• Is Part Of: ISRN geometry. Volume 2012(2012)
• Publication Date: 2012-06-10
• Subjects: Geometry -- Periodicals; Topology -- Periodicals; Geometry; Topology; Periodicals; 516
• DOI: 10.5402/2012/254235
• Languages: English
• ISSNs: 2090-6307

Abstract

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 H S 1 * (pt)-module basis for the S 1 -equivariant cohomology ring of the Hessenberg variety.