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This thesis is concerned with the geometry, topology and combinatorics arising in the study of arc algebras of types Bm-1 and Dm. These algebras are closely related to infinite-dimensional representation theory of Lie algebras, the geometry of perverse sheaves on isotropic Grassmannians and the representation theory of non-semisimple Brauer algebras. Results of Ehrig and Stroppel show that the center of the arc algebra of type Dm (which is isomorphic to the arc algebra of type Bm-1) is isomorphic to the cohomology ring of a certain two-row Springer fiber of type Dm.
In the first part of this thesis we combinatorially construct an explicit topological model for every two-row Springer fiber of type Dm and prove that the respective topological model is homeomorphic to its corresponding Springer fiber. In doing so, we confirm a conjecture by Ehrig and Stroppel concerning the topology of the equal-row Springer fiber for type Dm. Moreover, we show that every two-row Springer fiber of type Cm-1 is homeomorphic (even isomorphic as an algebraic variety) to a certain two-row Springer fiber of type Dm. This unexpected isomorphism of algebraic varieties can be interpreted as Langlands dual to the known isomorphism between the arc algebras of type Bm-1 and Dm.
In the second part we explain an elementary, topological construction of the Springer representation on the homology of (topological) Springer fibers of types Cm-1 and Dm in the two-row case. The Weyl group action and the component group action admit a diagrammatic description in terms of cup diagrams which appear in the definition of the arc algebras of types Bm-1 and Dm. We determine the decomposition of the representations into irreducibles and relate our construction to classical Springer theory. In addition to that, we give an explicit presentation of the cohomology rings of all two-row Springer fibers in types Cm-1 and Dm.
In the third part we describe a low-dimensional topology approach to understanding the arc algebras of types Bm-1 and Dm using two-dimensional surfaces and TQFTs. More precisely, we combinatorially describe the 2-category of singular cobordisms, called (rank one) foams, which governs the functorial version of (type A) Khovanov homology. As an application we use this singular cobordism construction to realize the arc algebras of type Bm-1 and Dm topologically by establishing an explicit isomorphism to a web algebra arising from foams. This result reduces the proof of the associativity of the arc algebras of type Bm-1 and Dm (which requires hard combinatorial work and a cumbersome amount of computations) to certain obvious topological equivalences between two-dimensional surfaces. Moreover, it shows how to remove the technical and unnatural condition of having to choose a certain admissible order of surgery moves in order for the multiplication to be well-defined in the original definition of the arc algebra.
