2010/12/18 by Darius Bayegan, Bayegan, Darius, Megumi Harada +1
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Primary: 14M17 #Secondary: 55N91
paper · pdf · doi:10.48550/arxiv.1012.4054
openalex publication_date 2010/12/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this manuscript we develop the theory of poset pinball, a combinatorial game recently introduced by Harada and Tymoczko for the study of the equivariant cohomology rings of GKM-compatible subspaces of GKM spaces. Harada and Tymoczko also prove that in certain circumstances, a successful outcome of Betti poset pinball yields a module basis for the equivariant cohomology ring of the GKM-compatible subspace. Our main contributions are twofold. First we construct an algorithm (which we call the dimension pair algorithm) which yields the result of 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 \Flags(\Cn). The definition of the algorithm is motivated by a correspondence between Hessenberg affine cells and certain Schubert polynomials which we learned from Erik Insko. Second, in the special case of the type A regular nilpotent Hessenberg varieties specified by the Hessenberg function h(1)=h(2)=3 and h(i) = i+1 for 3 ≤ i ≤ n-1 and h(n)=n, we prove that the pinball result coming from the dimension pair algorithm is poset-upper-triangular; by results of Harada and Tymoczko this implies the corresponding equivariant cohomology classes form a H^*S1(\pt)-module basis for the S1-equivariant cohomology ring of the Hessenberg variety.