Diagrams for perverse sheaves on isotropic Grassmannians and the supergroup SOSP(m|2n)

We describe diagrammatically a positively graded Koszul algebra \mathbb{D}_k such that the category of finite dimensional \mathbb{D}_k-modules is equivalent to the category of perverse sheaves on the isotropic Grassmannian of type D_k constructible with respect to the Schubert stratification. The connection is given by an explicit isomorphism to the endomorphism algebra of a projective generator described in by Braden. The algebra is obtained by a "folding" procedure from the generalized Khovanov arc algebras. We relate this algebra to the category of finite dimensional representations of the orthosymplectic supergroups. The proposed equivalence of categories gives a concrete description of the categories of finite dimensional SOSP(m|2n)-modules

Categorified sl(N) invariants of colored rational tangles

We use categorical skew Howe duality to find recursion rules that compute categorified sl(N) invariants of rational tangles colored by exterior powers of the standard representation. Further, we offer a geometric interpretation of these rules which suggests a connection to Floer theory. Along the way we make progress towards two conjectures about the colored HOMFLY homology of rational links.Comment: 45 pages, many figures, uses dcpic.sty, v2: minor changes and new example 5

Tensor product algebras, Grassmannians and Khovanov homology

We discuss a new perspective on Khovanov homology, using categorifications of tensor products. While in many ways more technically demanding than Khovanov's approach (and its extension by Bar-Natan), this has distinct advantage of directly connecting Khovanov homology to a categorification of \$(\mathbb{C}^2)^{\otimes \ell}\$, and admitting a direct generalization to other Lie algebras. While the construction discussed is a special case of that given in previous work of the author, this paper contains new results about the special case of \$\mathfrak{sl}_2\$ showing an explicit connection to Bar-Natan's approach to Khovanov homology, to the geometry of Grassmannians, and to the categorified Jones-Wenzl projectors of Cooper and Krushkal. In particular, we show that the colored Jones homology defined by our approach coincides with that of Cooper and Krushkal.Comment: v2: 37 pages. The paper has been extended at several points, and various small issues corrected following referee reports. Final published versio