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

Abstract

In this manuscript we study type A nilpotent Hessenberg varieties equipped with a natural S 1action 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 S 1 -fixed points of the nilpotent Hessenberg variety Hess(N, h) and the set of (h, λ N )-permissible fillings of the Young diagram λ N . Third, we use poset pinball, the combinatorial game introduced by Harada and Tymoczko, to study the S 1equivariant 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 Fℓags(C n ) ∼ = GL(n, C)/B which maps to a module basis of H * S 1 (S (n−2,2) ) under the projection map H * T (Fℓags(C n )) → H * S 1 (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 (S (n−2,2) ).We close with open questions for future work. CONTENTS 1. Introduction 1 2. Nilpotent Hessenberg varieties and S 1 -actions 3 3. Adjacent-pair matrices and highest forms of nilpotent operators 4 4. S 1 -fixed points in Hessenberg varieties and permissible fillings 12 5. Betti-acceptable pinball and linear independence 14 6. Small-n cases: n = 4 and n = 5 17 7. A poset pinball module basis for (n − 2, 2) Springer varieties 18 8. Open questions 24 References 24 1 We use English notation for Young diagrams. 2 We work with cohomology with coefficients in C throughout, and hence omit it from our notation.POSET PINBALL, HIGHEST FORMS, AND (n − 2, 2) SPRINGER VARIETIES 3 between permissible fillings and S 1 -fixed points of the Springer variety. The module basis is obtained by taking images under the natural projection map H * T (Fℓags(C n )) → H * S 1 (S (n−2,2) ), to be described in detail below, of a subset of the T -equivariant Schubert classes in H * T (Fℓags(C n )). A similar analysis by Bayegan and the second author in a special case of regular nilpotent Hessenberg varieties [2] yields a poset-uppertriangular basis in the sense of [9]. In contrast to the results in [2], in the present manuscript we find that the module basis is not poset-upper-triangular; this is t...

An equivariant basis for the cohomology of Springer fibers

“…Our Theorem 5.9 generalizes results of Harada-Tymoczko [22] and Dewitt-Harada [9] which address the case of λ = (n − 1, 1) and λ = (n − 2, 2), respectively. The main difficulty in generalizing the methods used in those papers is that the equivariant cohomology classes in H * S (B λ ) constructed via poset pinball may not satisfy upper triangular vanishing conditions (with respect to some partial ordering on the set of S-fixed points of B λ ).…”

“…We obtain an analogous statement for equivariant cohomology in Corollary 5.14. Our analysis generalizes work of Harada-Tymoczko [22] and Harada-Dewitt [9] in the sense that Corollary 5.14 implies the existence of an explicit module basis for H * S (B α ) constructed by playing poset pinball.…”

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

“…Thus our result gives rise to a new family of examples of Hessenberg varieties (and GKM-compatible subspaces) for which poset pinball successfully produces explicit module bases. We mention that the dimension pair algorithm also produces module bases in a special case of Springer varieties [7]. Although we do not know whether the dimension pair algorithm always succeeds in producing module bases for the S 1 -equivariant cohomology rings for a general nilpotent Hessenberg variety, the evidence thus far is suggestive.…”

“…In the case when N is principal nilpotent we take N hf = N since N is already in highest form and this is the form chosen by Tymoczko in [17]. A more detailed discussion of highest forms as it pertains to poset pinball theory is in [7].…”

“…The next proposition follows from the definitions and some linear algebra. It is proven and discussed in more detail in [7], where the notation used is slightly different.…”