Three-dimensional representations of braid groups associated with some finite complex reflection groups

摘要

We study the rigidity of three-dimensional representations of braid groups associated with finite primitive irreducible complex reflection groups in GL(3, ?). 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 Saito–Kato–Sekiguchi [Uniformization systems of equations with singularities along the discriminant sets of complex reflection groups of rank three, Kyushu J. Math. 68 (2014) 181–221; On the uniformization of complements of discriminant loci, RIMS Kokyuroku 287 (1977) 117–137]. Invariant Hermitian forms are also studied.