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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Complex specializations of the reduced Gassner representation of the pure braid group

Author: Mohammad N. Abdulrahim
Journal: Proc. Amer. Math. Soc. 125 (1997), 1617-1624
MSC (1991): Primary 20F36
MathSciNet review: 1422839
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Abstract: We will give a necessary and sufficient condition for the specialization of the reduced Gassner representation $G_{n}(z): P_{n} \to GL_{n-1}(\mathbb {C})$ to be irreducible. It will be shown that for $z=(z_{1},\ldots ,z_{n})\in (\mathbb {C}^{*})^{n}$, $G_{n}(z)$ is irreducible if and only if $z_{1}\ldots z_{n} \neq 1$.

References [Enhancements On Off] (What's this?)

  • [1] M. Abdulrahim, The Gassner Representation of the Pure Braid Group, Doctoral Thesis, The Pennsylvania State University, 1995.
  • [2] Joan S. Birman, Braids, links, and mapping class groups, Princeton University Press, Princeton, N.J., 1974. Annals of Mathematics Studies, No. 82. MR 0375281 (51 #11477)
    Joan S. Birman, Erratum: “Braids, links, and mapping class groups” (Ann. of Math. Studies, No. 82, Princeton Univ. Press, Princeton, N. J., 1974), Princeton University Press, Princeton, N. J., 1975. Based on lecture notes by James Cannon. MR 0425944 (54 #13894)
  • [3] Larry Dornhoff, Group representation theory. Part A: Ordinary representation theory, Marcel Dekker Inc., New York, 1971. Pure and Applied Mathematics, 7. MR 0347959 (50 #458a)
  • [4] E. Formanek, Braid group representations of low degree, Proc. London Math. Soc. 73 (1996), 279-322. CMP 96:15
  • [5] Vagn Lundsgaard Hansen, Braids and coverings: selected topics, London Mathematical Society Student Texts, vol. 18, Cambridge University Press, Cambridge, 1989. With appendices by Lars Gæde and Hugh R. Morton. MR 1247697 (94g:57004)

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Additional Information

Mohammad N. Abdulrahim
Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802

PII: S 0002-9939(97)04081-1
Keywords: Braid group, pure braid group, Burau representation, Gassner representation
Received by editor(s): December 23, 1995
Additional Notes: The results in this paper were written under the direction of Professor Edward Formanek whose help and encouragement are greatly appreciated.
Communicated by: Ronald M. Solomon
Article copyright: © Copyright 1997 American Mathematical Society