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Highest weight categories arising from Khovanov’s diagram algebra III: category $\mathcal {O}$

Authors: Jonathan Brundan and Catharina Stroppel
Journal: Represent. Theory 15 (2011), 170-243
MSC (2010): Primary 17B10, 16S37
Published electronically: March 7, 2011
MathSciNet review: 2781018
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Abstract: We prove that integral blocks of parabolic category $\mathcal {O}$ associated to the subalgebra $\mathfrak {gl}_m(\mathbb {C}) \oplus \mathfrak {gl}_n(\mathbb {C})$ of $\mathfrak {gl}_{m+n}(\mathbb {C})$ are Morita equivalent to quasi-hereditary covers of generalised Khovanov algebras. Although this result is in principle known, the existing proof is quite indirect, going via perverse sheaves on Grassmannians. Our new approach is completely algebraic, exploiting Schur-Weyl duality for higher levels. As a by-product we get a concrete combinatorial construction of $2$-Kac-Moody representations in the sense of Rouquier corresponding to level two weights in finite type $A$.

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Additional Information

Jonathan Brundan
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403

Catharina Stroppel
Affiliation: Department of Mathematics, University of Bonn, 53115 Bonn, Germany

Received by editor(s): July 15, 2009
Received by editor(s) in revised form: June 22, 2010, and June 26, 2010
Published electronically: March 7, 2011
Additional Notes: The first author was supported in part by NSF grant no. DMS-0654147
The second author was supported by the NSF and the Minerva Research Foundation DMS-0635607.
Article copyright: © Copyright 2011 American Mathematical Society