2-row Springer Fibres and Khovanov Diagram Algebras for Type D

We study in detail two row Springer fibres of even orthogonal type from an algebraic as well as a topological point of view. We show that the irreducible components and their pairwise intersections are iterated ${{\mathbb{P}}^{1}}$ -bundles. Using results of Kumar and Procesi we compute the cohomolo...

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Veröffentlicht in:Canadian journal of mathematics 2016-12, Vol.68 (6), p.1285-1333
Hauptverfasser: Ehrig, Michael, Stroppel, Catharina
Format: Artikel
Sprache:eng
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Zusammenfassung:We study in detail two row Springer fibres of even orthogonal type from an algebraic as well as a topological point of view. We show that the irreducible components and their pairwise intersections are iterated ${{\mathbb{P}}^{1}}$ -bundles. Using results of Kumar and Procesi we compute the cohomology ring with its action of the Weyl group. The main tool is a type $\text{D}$ diagram calculus labelling the irreducible components in a convenient way that relates to a diagrammatical algebra describing the category of perverse sheaves on isotropic Grassmannians based on work of Braden. The diagram calculus generalizes Khovanov's arc algebra to the type $\text{D}$ setting and should be seen as setting the framework for generalizing well-known connections of these algebras in type $\text{A}$ to other types.
ISSN:0008-414X
1496-4279
DOI:10.4153/CJM-2015-051-4