The Long-Moody construction and polynomial functors

Arthur Soulié (1, 2)

1 IRMA - Institut de Recherche Mathématique Avancée
2 University of Glasgow

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.

Mots clés: Functor categories; Burau representations; Long-Moody Construction; Braid groups; braid groups; functor categories; Long-Moody construction; polynomial functors

Domaines: Topologie algébrique [math.AT]; Catégories et ensembles [math.CT]

Arthur Soulié. The Long-Moody construction and polynomial functors. Annales de l'Institut Fourier, 2019, 69 (4), pp.1799-1856. ⟨10.5802/aif.3282⟩. ⟨hal-01470761v6⟩