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Invariant subspaces of the Lawrence-Krammer representation


Author: Claire Levaillant
Journal: Proc. Amer. Math. Soc. 140 (2012), 2599-2612
MSC (2010): Primary 20F36; Secondary 20C08
DOI: https://doi.org/10.1090/S0002-9939-2011-11107-9
Published electronically: November 30, 2011
MathSciNet review: 2910748
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Abstract: The Lawrence-Krammer representation was used in $ 2000$ to show the linearity of the braid group. The problem had remained open for many years. The fact that the Lawrence-Krammer representation of the braid group is reducible for some complex values of its two parameters is now known, as well as the complete description of these values. It is also known that when the representation is reducible, the action on a proper invariant subspace is an Iwahori-Hecke algebra action. In this paper, we prove a theorem of classification for the invariant subspaces of the Lawrence-Krammer space. We classify the invariant subspaces in terms of Specht modules. We fully describe them in terms of dimension and spanning vectors in the Lawrence-Krammer space.


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Additional Information

Claire Levaillant
Affiliation: Department of Mathematics, Caltech, Pasadena, California 91125
Address at time of publication: Department of Mathematics, University of California Santa Barbara, Santa Barbara, California 93106
Email: cl@caltech.edu, claire@math.ucsb.edu

DOI: https://doi.org/10.1090/S0002-9939-2011-11107-9
Keywords: Braid groups, representation theory
Received by editor(s): July 31, 2010
Received by editor(s) in revised form: January 28, 2011, February 15, 2011, and March 4, 2011
Published electronically: November 30, 2011
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2011 American Mathematical Society

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