The Long-Moody construction and polynomial functors

Abstract

In 1994, Long and Moody gave a construction on representations of braid groups which associates a representation of \mathbf{B}\_{n} with a representation of \mathbf{B}\_{n+1}. In this paper, we prove that this construction is functorial and can be extended: it inspires endofunctors, called Long-Moody functors, between 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.