The Long–Moody construction and polynomial functors

In 1994, Long and Moody gave a construction on representations of braid groups which associates a representation of Bn with a representation of Bn+1. In this paper, we prove that this construction is functorial and can be extended: it inspires endofunctors, called Long–Moody functors, on the category of functors from Quillen’s bracket construction associated with the braid groupoid to a module category. Then we study the effect of Long–Moody functors on strong polynomial functors: we prove that they increase by one the degree of very strong polynomiality