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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A faithfulness criterion for the Gassner representation of the pure braid group
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by Mohammad N. Abdulrahim PDF
Proc. Amer. Math. Soc. 125 (1997), 1249-1257 Request permission

Abstract:

This work is directed towards the open question of the faithfulness of the reduced Gassner representation of the pure braid group, $P_{n}(n > 3)$. Long and Paton proved that if a Burau matrix $M$ has ones on the diagonal and zeros below the diagonal then $M$ is the identity matrix. In this paper, a generalization of Long and Paton’s result will be proved. Our main theorem is that if the trace of the image of an element of $P_{n}$ under the reduced Gassner representation is $n-1$, then this element lies in the kernel of this representation. Then, as a corollary, we prove that an analogue of the main theorem holds true for the Burau representation of the braid group.
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Additional Information
  • Mohammad N. Abdulrahim
  • Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802
  • Email: mna@math.psu.edu
  • Received by editor(s): October 31, 1995
  • Additional Notes: The results in this paper are written under the direction of Professor Edward Formanek whose help and encouragement are greatly appreciated.
  • Communicated by: Ronald M. Solomon
  • © Copyright 1997 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 125 (1997), 1249-1257
  • MSC (1991): Primary 20F36
  • DOI: https://doi.org/10.1090/S0002-9939-97-03827-6
  • MathSciNet review: 1389500