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

 

Low-dimensional unitary representations of $B_3$


Author: Imre Tuba
Journal: Proc. Amer. Math. Soc. 129 (2001), 2597-2606
MSC (1991): Primary 20F36, 20C07, 81R10; Secondary 20H20, 16S34
Published electronically: March 15, 2001
MathSciNet review: 1838782
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Abstract:

We characterize all simple unitarizable representations of the braid group $B_3$on complex vector spaces of dimension $d \leq 5$. In particular, we prove that if $\sigma_1$ and $\sigma_2$ denote the two generating twists of $B_3$, then a simple representation $\rho:B_3 \to \operatorname{GL} (V)$ (for $\dim V \leq 5$) is unitarizable if and only if the eigenvalues $\lambda_1, \lambda_2, \ldots, \lambda_d$ of $\rho(\sigma_1)$ are distinct, satisfy $\vert\lambda_i\vert=1$ and $\mu^{(d)}_{1i} > 0$ for $2 \leq i \leq d$, where the $\mu^{(d)}_{1i}$ are functions of the eigenvalues, explicitly described in this paper.


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Additional Information

Imre Tuba
Affiliation: Department of Mathematics, Mail Code 0112, University of California, San Diego, 9500 Gilman Dr., La Jolla, California 92093-0112
Address at time of publication: Department of Mathematics, University of California, Santa Barbara, California 93106
Email: ituba@math.ucsd.edu, ituba@math.ucsb.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-01-05903-2
PII: S 0002-9939(01)05903-2
Received by editor(s): August 31, 1999
Received by editor(s) in revised form: January 31, 2000
Published electronically: March 15, 2001
Communicated by: Stephen D. Smith
Article copyright: © Copyright 2001 American Mathematical Society