# Algebra/Topology Seminar by Arthur Soulie – University of Copenhagen

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# Algebra/Topology Seminar by Arthur Soulie

Talk by Arthur Soulie (University of Strasbourg)

Title: Action of the Long-Moody Construction on Polynomial Functors
Abstract: In 2015, Randal-Williams and Wahl proved homological stability with certain twisted coefficients for dif- ferent families of groups, in particular the one of braid groups. In fact, they obtain the stability for coefficients given by functors satisfying polynomial conditions. We only know few examples of such functors. Among them, we have the functor given by the unreduced Burau representations. 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. This construction complexifies in a sense the initial representation: for instance, starting from a dimension one representation, one obtains the unreduced Burau representation. In this talk, I will present this construction from a functorial point of view. I will explain that the construction of Long and Moody defines an endofunctor, called the Long-Moody functor, between a suitable category of functors. Then, after defining strong polynomial functors in this context, I will prove that the Long-Moody functor increases by one the degree of strong polynomiality of a strong polynomial functor. Thus, the Long-Moody construction will provide new examples of twisted coefficients corresponding to the framework of Randal-Williams and Wahl.