Poset pinball, GKM-compatible subspaces, and Hessenberg varieties

Abstract

This paper has three main goals. First, we set up a general framework to address the problem of constructing module bases for the equivariant cohomology of certain subspaces of GKM spaces. To this end we introduce the notion of a GKM-compatible subspace of an ambient GKM space. We also discuss poset-upper-triangularity, a key combinatorial notion in both GKM theory and more generally in localization theory in equivariant cohomology. With a view toward other applications, we present parts of our setup in a general algebraic and combinatorial framework. Second, motivated by our central problem of building module bases, we introduce a combinatorial game which we dub poset pinball and illustrate with several examples. Finally, as first applications, we apply the perspective of GKM-compatible subspaces and poset pinball to construct explicit and computationally convenient module bases for the $S^1$-equivariant cohomology of all Peterson varieties of classical Lie type, and subregular Springer varieties of Lie type $A$. In addition, in the Springer case we use our module basis to lift the classical Springer representation on the ordinary cohomology of subregular Springer varieties to $S^1$-equivariant cohomology in Lie type $A$.

32 pages, 4 figures

References in corpus (4)

• A positive Monk formula in the S^1-equivariant cohomology of type A Peterson varieties
• Schubert Classes in the Equivariant K-Theory and Equivariant Cohomology of the Grassmannian
• On the equivariant cohomology of subvarieties of a B-regular variety
• Towards Generalizing Schubert Calculus in the Symplectic Category

Cited by in corpus (9)

• The equivariant cohomology rings of Peterson varieties in all Lie types
• The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A
• The equivariant cohomology rings of regular nilpotent Hessenberg varieties in Lie type A: a research announcement
• Affine pavings of regular nilpotent Hessenberg varieties and intersection theory of the Peterson variety
• A Giambelli formula for the $S^1$-equivariant cohomology of type A Peterson varieties
• The $S^1$-equivariant cohomology rings of $(n-k,k)$ Springer varieties
• Solomon-Terao algebra of hyperplane arrangements
• Poset pinball, the dimension pair algorithm, and type A regular nilpotent Hessenberg varieties
• Poset pinball, highest forms, and (n-2,2) Springer varieties