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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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Some homological properties of the category $\mathcal {O}$, II
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by Volodymyr Mazorchuk PDF
Represent. Theory 14 (2010), 249-263 Request permission


We show, in full generality, that Lusztig’s $\mathbf {a}$-function describes the projective dimension of both indecomposable tilting modules and indecomposable injective modules in the regular block of the BGG category $\mathcal {O}$, proving a conjecture from the first paper. On the way we show that the images of simple modules under projective functors can be represented in the derived category by linear complexes of tilting modules. These complexes, in turn, can be interpreted as the images of simple modules under projective functors in the Koszul dual of the category $\mathcal {O}$. Finally, we describe the dominant projective modules and also the projective-injective modules in some subcategories of $\mathcal {O}$ and show how one can use categorification to decompose the regular representation of the Weyl group into a direct sum of cell modules, extending the results known for the symmetric group (type $A$).
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Additional Information
  • Volodymyr Mazorchuk
  • Affiliation: Department of Mathematics, Uppsala University, SE-751 06, Uppsala, Sweden
  • MR Author ID: 353912
  • Email:
  • Received by editor(s): September 15, 2009
  • Received by editor(s) in revised form: October 3, 2009
  • Published electronically: March 1, 2010
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Represent. Theory 14 (2010), 249-263
  • MSC (2000): Primary 16E10, 16E30, 16G99, 17B10
  • DOI:
  • MathSciNet review: 2602033