Poset pinball, highest forms, and (n-2,2) Springer varieties
Abstract
We study type A nilpotent Hessenberg varieties equipped with a natural S1-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 λ= (n-2,2) for n ≥ 4. First we define the adjacent-pair matrix corresponding to any filling of a Young diagram with n boxes with the alphabet \1,2,…,n\. 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 N and Hessenberg function h, we construct an explicit bijection between the S1-fixed points of the nilpotent Hessenberg variety (N,h) and the set of (h,λN)-permissible fillings of the Young diagram λN. Third, we use poset pinball, the combinatorial game introduced by Harada and Tymoczko, to study the S1-equivariant cohomology of type A Springer varieties S(n-2,2) associated to Young diagrams of shape (n-2,2) for n≥ 4. 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 T-equivariant cohomology of the flag variety F ags(n) which maps to a module basis of H*S1(S(n-2,2)) under the projection H*T(F ags(n)) H*S1(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 H*S1(S(n-2,2)). We close with open questions for future work.