Poset pinball, highest forms, and (n-2,2) Springer varieties
arXiv:1012.5265
Abstract
We study type nilpotent Hessenberg varieties equipped with a natural -action using techniques introduced by Tymoczko, Harada-Tymoczko, and Bayegan-Harada, with a particular emphasis on a special class of nilpotent Springer varieties corresponding to the partition for . First we define the adjacent-pair matrix corresponding to any filling of a Young diagram with boxes with the alphabet . Using the adjacent-pair matrix we make more explicit and also extend some statements concerning highest forms of linear operators in previous work of Tymoczko. Second, for a nilpotent operator and Hessenberg function , we construct an explicit bijection between the -fixed points of the nilpotent Hessenberg variety $\Hess(N,h)$ and the set of -permissible fillings of the Young diagram . Third, we use poset pinball, the combinatorial game introduced by Harada and Tymoczko, to study the -equivariant cohomology of type Springer varieties associated to Young diagrams of shape for . Specifically, we use the dimension pair algorithm for Betti-acceptable pinball described by Bayegan and Harada to specify a subset of the equivariant Schubert classes in the -equivariant cohomology of the flag variety $\mathcal{F}\ell ags(\C^n)$ which maps to a module basis of under the projection $H^*_T(\mathcal{F}\ell ags(\C^n)) \to H^*_{S^1}(\mathcal{S}_{(n-2,2)})$. Our pinball module basis is not poset-upper-triangular; this is the first concrete such example in the literature. A consequence of our proof is that there exists a simple and explicit change of basis which transforms our basis to a poset-upper-triangular module basis for . We close with open questions for future work.
25 pages, minor changes in exposition, typos corrected