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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Highest weight categories arising from Khovanov’s diagram algebra III: category $\mathcal {O}$
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by Jonathan Brundan and Catharina Stroppel PDF
Represent. Theory 15 (2011), 170-243 Request permission

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
  • Email: brundan@uoregon.edu
  • Catharina Stroppel
  • Affiliation: Department of Mathematics, University of Bonn, 53115 Bonn, Germany
  • Email: stroppel@math.uni-bonn.de
  • 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.
  • © Copyright 2011 American Mathematical Society
  • Journal: Represent. Theory 15 (2011), 170-243
  • MSC (2010): Primary 17B10, 16S37
  • DOI: https://doi.org/10.1090/S1088-4165-2011-00389-7
  • MathSciNet review: 2781018