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ISRN Algebra
Volume 2012 (2012), Article ID 858959, 13 pages
Research Article

Another Proof of the Faithfulness of the Lawrence-Krammer Representation of the Braid Group 𝐡 πŸ‘

Department of Mathematics, Beirut Arab University, P.O. Box 11-5020, Beirut 11072809, Lebanon

Received 16 March 2012; Accepted 6 May 2012

Academic Editors: P. Koshlukov, H. Li, S. Yang, and Y. Zhou

Copyright © 2012 Mohammad N. Abdulrahim and Mariam Hariri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The Lawrence-Krammer representation of the braid group 𝐡 𝑛 was proved to be faithful for 𝑛 β‰₯ 3 by Bigelow and Krammer. In our paper, we give a new proof in the case 𝑛 = 3 by using matrix computations. First, we prove that the representation of the braid group 𝐡 3 is unitary relative to a positive definite Hermitian form. Then we show the faithfulness of the representation by specializing the indeterminates q and t to complex numbers on the unit circle rather than specializing them to real numbers as what was done by Krammer.