Low-Dimensional Unitary Representations of B_3

Details Low-Dimensional Unitary Representations of B_3 Low-Dimensional Unitary Representations of B_3

1999-08-20

arxiv.org/abs/math/9908110v3

Mathematics

Representation Theory

Added sections + some minor updates

Scientific paper

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 $\rho:B_3 \to \gl(V)$ (for $\dim V \leq 5$) is unitarizable if and only if the eigenvalues $\lambda_1, \lambda_2, ..., \lambda_d$ of $\rho(\sigma_1)$ are distinct, satisfy $|\lambda_i|=1$ 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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