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

Abstract

We study the rigidity of three-dimensional representations of braid groups associated with finite primitive irreducible complex reflection groups in [Formula: see text]. 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.