Parabolic category O, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology

By C. Stroppel. Compositio Mathematica, 2006

132 citations recorded by the source corpus.

Abstract For a fixed parabolic subalgebra 𝔭 of $\mathfrak {gl}(n,\mathbb {C})$ we prove that the centre of the principal block 𝒪0𝔭 of the parabolic category 𝒪 is naturally isomorphic to the cohomology ring H*(ℬ𝔭) of the corresponding Springer fibre. We give a diagrammatic description of 𝒪0𝔭 for maximal parabolic 𝔭 and give an explicit isomorphism to Braden’s description of the category PervB(G(k,n)) of Schubert-constructible perverse sheaves on Grassmannians. As a consequence Khovanov’s algebra ℋn is realised as the endomorphism ring of some object from PervB(G(n,n)) which corresponds under localisation and the Riemann–Hilbert correspondence to a full projective–injective module in the corresponding category 𝒪0𝔭. From there one can deduce that Khovanov’s tangle invariants are obtained from the more general functorial invariants in [C. Stroppel, Categorification of the Temperley Lieb category, tangles, and cobordisms via projective functors, Duke Math. J. 126(3) (2005), 547–596] by restriction.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0483443dc597c556ebcc458c5d109d45d86adee6

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Buildings, spiders, and geometric Satake (2011, methodology): “This Springer fibre formalism and this description of it have been used as a model of Khovanov homology [22,41].”
Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m) (2012, methodology): “More generally, for link homology theories associated with fundamental sln representations, Webster describes an isomorphism relating his construction to Sussan’s category O based link homology theory [69] , which is related via Koszul duality to a theory defined by Mazorchuk and Stroppel [56].”
Gale duality and Koszul duality (2008, result): “We prove the following three facts about the rings A and B, all of which are analogous to results about categoryO and the geometry of the Springer resolution [Spa76, Irv85, Bru08, Stra , SW].”
A combinatorial approach to functorial quantum slk knot invariants (2007, background): “In light of [7] and [29] one might expect that not only the partial flag varieties, but also Springer fibres and Spaltenstein varieties, and the combinatorics of their cohomology rings should play a crucial role in the complete picture.”