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Poset pinball, the dimension pair algorithm, and type \(A\) regular nilpotent Hessenberg varieties. (English) Zbl 1276.14070

In the paper under review, the authors develop the theory of poset pinball, a combinatorial game introduced in [M. Harada and J. Tymoczko, “Poset pinball, GKM-compatible subspaces, and Hessenberg varieties”, arXiv:1007.2750] to study the equivariant cohomology ring of a GKM-compatible subspace \(X\) of a GKM space (see Definition 4.5 of [loc. cit.]); Harada and Tymoczko also proved that, in certain circumstances, a successful outcome of Betti poset pinball yields a module basis for the equivariant cohomology ring of \(X\). In this paper, the authors first define a dimension pair algorithm, which yields a successful outcome of Betti poset pinball for any type \(A\) regular nilpotent Hessenberg and any type \(A\) nilpotent Springer variety, considered as GKM-compatible subspaces of the flag variety. This algorithm is motivated by a correspondence between Hessenberg affine cells and certain Schubert polynomials. Second, in a special case of regular nilpotent Hessenberg varieties, the authors prove that the pinball outcome is poset-upper-triangular, and hence the corresponding classes form a \(H^\ast_{S^1}\)(pt)-module basis for the \(S^{1}\)-equivariant cohomology ring of the Hessenberg variety.

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
05E10 Combinatorial aspects of representation theory
55N91 Equivariant homology and cohomology in algebraic topology
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
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References:

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