2-row Springer fibres and Khovanov diagram algebras for type D

TL;DR

This paper explores two-row Springer fibres of even orthogonal type, revealing their geometric structure, computing their cohomology, and developing a diagram calculus that generalizes Khovanov's arc algebra to type D, linking algebraic and topological perspectives.

Contribution

It introduces a type D diagram calculus that generalizes Khovanov's arc algebra, connecting Springer fibres, perverse sheaves, and Brauer algebras in a novel way.

Findings

Irreducible components are iterated P^1-bundles.

Cohomology ring computed with Weyl group action.

Diagram calculus relates to perverse sheaves on Grassmannians.

Abstract

We study in detail two row Springer fibres of even orthogonal type from an algebraic as well as topological point of view. We show that the irreducible components and their pairwise intersections are iterated 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 D diagram calculus labelling the irreducible components in an convenient way which 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 D setting and should be seen as setting the framework for generalizing well-known connections of these algebras in type A to other types. The results will be connected to Brauer algebras at non-generic parameters in a subsequent paper.