Mathematics – Representation Theory
Scientific paper
1999-12-02
Mathematics
Representation Theory
To appear in the Pacific Journal of Mathematics
Scientific paper
We give a complete classification of simple representations of the braid group B_3 with dimension $\leq 5$ over any algebraically closed f ield. In particular, we prove that a simple d-dimensional representation $\rho: B_3 \to GL(V)$ is determined up to isomorphism by the eigenvalues $\lambda_1, \lambda_2, ..., \lambda_d$ of the image of the generators for d=2,3 and a choice of a $\delta=\sqrt{\det \rho(\sigma_1)}$ for d=4 or a choice of $\delta=\sqrt[5]{\det \rho(\sigma_1)}$ for d=5. We also s howed that such representations exist whenever the eigenvalues and $\delta$ are not roots of certain polynomials $Q_{ij}^{(d)}$, which are explicitly given. In this case, we construct the matrices via which the generators act on V. As an application of our techniques, we also obtain nontrivial q-versions of some of Deligne's formulas for dimensions of representations of exceptional Lie groups.
Tuba Imre
Wenzl Hans
No associations
LandOfFree
Representations of the braid group B_3 and of SL(2,Z) does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Representations of the braid group B_3 and of SL(2,Z), we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Representations of the braid group B_3 and of SL(2,Z) will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-262262