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July, 2017 Poset pinball, GKM-compatible subspaces, and Hessenberg varieties
Megumi HARADA, Julianna TYMOCZKO
J. Math. Soc. Japan 69(3): 945-994 (July, 2017). DOI: 10.2969/jmsj/06930945


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 $A$.


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Megumi HARADA. Julianna TYMOCZKO. "Poset pinball, GKM-compatible subspaces, and Hessenberg varieties." J. Math. Soc. Japan 69 (3) 945 - 994, July, 2017.


Published: July, 2017
First available in Project Euclid: 12 July 2017

zbMATH: 1384.55005
MathSciNet: MR3685032
Digital Object Identifier: 10.2969/jmsj/06930945

Primary: 55N91
Secondary: 14L30 , 22E46

Keywords: equivariant cohomology and localization , Goresky–Kottwitz–MacPherson theory , graded partially ordered sets , nilpotent Hessenberg varieties , Springer theory

Rights: Copyright © 2017 Mathematical Society of Japan


Vol.69 • No. 3 • July, 2017
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