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Representations of the braid group B_3 and of SL(2,Z) | | Imre Tuba
; Hans Wenzl
; | | Date: |
2 Dec 1999 | | Subject: | Representation Theory; Group Theory; Rings and Algebras; Quantum Algebra MSC-class: 20F36, 20C07, 81R10 (Primary); 16S34, 15A69 (Secondary) | math.RT math.GR math.QA math.RA | | Abstract: | 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 $
ho: B_3 o 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 $delta=sqrt{det
ho(sigma_1)}$ for d=4 or a choice of $delta=sqrt[5]{det
ho(sigma_1)}$ 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. | | Source: | arXiv, math.RT/9912013 | | Services: | Forum | Review | PDF | Favorites | |
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