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

In this manuscript we study type $A$ nilpotent Hessenberg varieties equipped with a natural $S^1$-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 $\lambda= (n-2,2)$ for $n \geq 4$. First we define the adjacent-pair matrix corresponding to any filling of a Young diagram with $n$ boxes with the alphabet $\{1,2,\ldots,n\}$. 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 $S^1$-fixed points of the nilpotent Hessenberg variety Hess$(N,h)$ and the set of $(h,\lambda_N)$-permissible fillings of the Young diagram $\lambda_N$. Third, we use poset pinball, the combinatorial game introduced by Harada and Tymoczko, to study the $S^1$-equivariant cohomology of type $A$ Springer varieties $\mathcal{S}_{(n-2,2)}$ associated to Young diagrams of shape $(n-2,2)$ for $n\geq 4$. 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 $\mathbb{T}$-equivariant cohomology of the flag variety $\mathcal{F}\ell ags(\mathbb{C}^n) \cong GL(n,\mathbb{C})/B$ which maps to a module basis of $H^*_{S^1}(\mathcal{S}_{(n-2,2)})$ under the projection map $H^*_\mathbb{T}(\mathcal{F}\ell ags(\mathbb{C}^n)) \to H^*_{S^1}(\mathcal{S}_{(n-2,2)})$. 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 poset-upper-triangular module basis for $H^*_{S^1}(\mathcal{S}_{(n-2,2)})$. We close with open questions for future work.