We consider the general problem of establishing irreducibility criteria for the tensor product of two irreducible representations of a fundamental group $G=\pi _1(X)$, in particular when $X$ is the complement of hypersurfaces in a projective space. We set up an ad-hoc formalism and use a monodromy approach to define a of irreducible representations of $G$ whose tensor products remain irreducible for generic values of defining parameters. This is applied to the pure braid group, and yields the result that the action of the pure braid group is irreducible on the tensor products of a wide of representations (for generic parameters). The family of representations concerned here includes the representations of the Hecke algebras of type $A$, of the Birman-Wenzl-Murakami algebra, and the Yang-Baxter actions on the tensor products of $\mathfrak {sl}_2(\mathbb {C})$-modules. We then also apply this to the Hecke algebra representations of generalized braid groups. Finally, we define and get results on “infinitesimal Hecke algebras”, which are convenient objects to study tensor products decompositions of Hecke algebra representations. In particular, we show that not only the alternating powers, but every Schur functor applied to the reflection representation of Hecke algebras yield irreducible representations of the corresponding pure braid group.