Table of Contents
ISRN Geometry
Volume 2012, Article ID 254235, 34 pages
http://dx.doi.org/10.5402/2012/254235
Research Article

Poset Pinball, the Dimension Pair Algorithm, and Type 𝐴 Regular Nilpotent Hessenberg Varieties

Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada L8S 4K1

Received 28 January 2012; Accepted 15 March 2012

Academic Editors: L. C. Jeffrey, A. Morozov, and E. H. Saidi

Copyright Β© 2012 Darius Bayegan and Megumi Harada. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 π»βˆ—π‘†1(pt)-module basis for the S1-equivariant cohomology ring of the Hessenberg variety.