Poset Pinball, Highest Forms, and (n-2, 2) Springer Varieties

In this manuscript 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 Hess(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(Cn) ∼= GL(n,C)/B which maps to a module basis ofH∗ S1(S(n−2,2)) under the projection map H∗ T(F`ags(C))→ H∗ S1(S(n−2,2)). Our poset pinball module basis is not poset-upper-triangular; this is the first concrete such example in the literature. A straightforward consequence of our proof is that there exists a simple and explicit change of basis which transforms our poset pinball basis to a posetupper-triangular module basis for H∗ S1(S(n−2,2)). We close with open questions for future work. ∗Partially supported by an NSERC Discovery Grant, an NSERC University Faculty Award, and an Ontario Ministry of Research and Innovation Early Researcher Award. the electronic journal of combinatorics 19 (2012), #P56 1