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Three-dimensional representations of braid groups associated with some finite complex reflection groups. (English) Zbl 1469.32009

Summary: We study the rigidity of three-dimensional representations of braid groups associated with finite primitive irreducible complex reflection groups in \(\mathrm{GL}(3, \mathbb C)\). In many cases, we show the rigidity. For rigid representations, we give explicit forms of the representations, which turns out to be the monodromy representations of uniformization equations of M. Kato and J. Sekiguchi [Kyushu J. Math. 68, No. 1, 181–221 (2014; Zbl 1292.35007); K. Saito, “On the uniformization of complements of discriminant loci”, RIMS Kôkyûroku 287, 117–137 (1977), http://hdl.handle.net/2433/106125]. Invariant Hermitian forms are also studied.

MSC:

32G34 Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation)
14N20 Configurations and arrangements of linear subspaces
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
58A17 Pfaffian systems

Citations:

Zbl 1292.35007
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References:

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