2010/07/16 by Megumi Harada, Harada, Megumi, Julianna Tymoczko +1 · 2 citations
Mathematics · #14L30 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Primary: 55N91 #Secondary: 22E46
paper · pdf · doi:10.48550/arxiv.1007.2750
openalex publication_date 2010/07/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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 S1-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 S1-equivariant cohomology in Lie type A.