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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Schur functors and dominant dimension


Authors: Ming Fang and Steffen Koenig
Journal: Trans. Amer. Math. Soc. 363 (2011), 1555-1576
MSC (2010): Primary 16G10, 13E10
Published electronically: October 15, 2010
MathSciNet review: 2737277
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Abstract: The dominant dimension of an algebra $ A$ provides information about the connection between $ A\textrm{-mod}$ and $ B\textrm{-mod}$ for $ B=eAe$, a certain centralizer subalgebra of $ A$. Well-known examples of such a situation are the connection (given by Schur-Weyl duality) between Schur algebras and group algebras of symmetric groups, and the connection (given by Soergel's 'Struktursatz') between blocks of the category $ \mathcal O$ of a complex semisimple Lie algebra and the coinvariant algebra. We study cohomological aspects of such connections, in the framework of highest weight categories. In this setup we characterize the dominant dimension of $ A$ by the vanishing of certain extension groups over $ A$, we determine the range of degrees, for which certain cohomology groups over $ A$ and over $ eAe$ get identified, we show that Ringel duality does not change dominant dimensions and we determine the dominant dimension of Schur algebras.


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Additional Information

Ming Fang
Affiliation: Institute of Mathematics, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
Email: fming@amss.ac.cn

Steffen Koenig
Affiliation: Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany
Address at time of publication: Institut für Algebra und Zahlentheorie, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
Email: skoenig@math.uni-koeln.de

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05177-3
PII: S 0002-9947(2010)05177-3
Keywords: Schur functor, dominant dimension, quasi-hereditary algebra
Received by editor(s): December 3, 2008
Received by editor(s) in revised form: July 28, 2009
Published electronically: October 15, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.