Poset pinball, GKM-compatible subspaces, and Hessenberg varieties

TL;DR

This paper develops a framework for constructing module bases in equivariant cohomology of GKM-compatible subspaces, introduces a combinatorial game called poset pinball, and applies these to Peterson and Springer varieties, enabling explicit computations and representation lifts.

Contribution

It introduces the notion of GKM-compatible subspaces, the poset pinball game, and applies these to explicitly construct module bases for Peterson and Springer varieties in Lie type A.

Findings

Constructed explicit module bases for Peterson varieties.

Built module bases for subregular Springer varieties in Lie type A.

Lifted Springer representations to equivariant cohomology using the new bases.

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 $A$.