# The Long-Moody construction and polynomial functors

## TL;DR

This paper explores the functorial properties of the Long-Moody construction, extending it to endofunctors on polynomial functors, and shows it increases the degree of polynomiality by one.

## Contribution

It proves the functoriality of the Long-Moody construction and its extension to endofunctors, revealing how it affects polynomial functors in the braid group context.

## Key findings

- Long-Moody functors are functorial and extendable.
- They increase the degree of strong polynomiality by one.
- The construction links braid group representations with 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.