Poset pinball, GKM-compatible subspaces, and Hessenberg varieties

摘要

This paper has three main goals. First, we set up a general framework toaddress the problem of constructing module bases for the equivariant cohomologyof certain subspaces of GKM spaces. To this end we introduce the notion of aGKM-compatible subspace of an ambient GKM space. We also discussposet-upper-triangularity, a key combinatorial notion in both GKM theory andmore generally in localization theory in equivariant cohomology. With a viewtoward other applications, we present parts of our setup in a general algebraicand combinatorial framework. Second, motivated by our central problem ofbuilding module bases, we introduce a combinatorial game which we dub posetpinball and illustrate with several examples. Finally, as first applications,we apply the perspective of GKM-compatible subspaces and poset pinball toconstruct 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 Springercase we use our module basis to lift the classical Springer representation onthe ordinary cohomology of subregular Springer varieties to $S^1$-equivariantcohomology in Lie type $A$.