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arXiv math/9912013math.RTmath.GRmath.QAmath.RA
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辫群B_3和SL（2，Z）的表示

Representations of the braid group B_3 and of SL(2,Z)

Imre Tuba, Hans Wenzl

备注：
To appear in the Pacific Journal of Mathematics
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摘要：

给出了代数闭域上维数为$\leq 5$的辫群B_3的单表示的完全分类.特别是，我们证明了一个简单的d维表示$ρ: B_3 \to GL(V)$确定同构的特征值$λ_1, λ_2, ..., λ_d$的图像的生成器为d= 2，3和选择的$δ=\sqrt{\det ρ(σ_1)}$为d=4或选择的$δ=\sqrt[5]{\det ρ(σ_1)}$为d=5。当特征值和$δ$不是显式给出的某些多项式$Q_{ij}^{(d)}$的根时，也存在这样的表示.在这种情况下，我们构建的矩阵通过该发电机作用于V.作为我们的技术的应用，我们还获得了非平凡的Q版本的Deligne的一些公式的尺寸表示的例外李群。

英文摘要：

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 $ρ: B_3 \to GL(V)$ is determined up to isomorphism by the eigenvalues $λ_1, λ_2, ..., λ_d$ of the image of the generators for d=2,3 and a choice of a $δ=\sqrt{\det ρ(σ_1)}$ for d=4 or a choice of $δ=\sqrt[5]{\det ρ(σ_1)}$ for d=5. We also s howed that such representations exist whenever the eigenvalues and $δ$ 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.

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