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 B n with a representation of 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.

Twisted homological stability for configuration spaces

“…Another example of a finite-degree twisted coefficient system on Uβ is constructed in [Sou17, Proposition 2.29] from the Tong-Yang-Ma representations [TYM96]. Moreover, the main result of [Sou17] is a functorial version of the Long-Moody construction [Lon94], which produces a new twisted coefficient system on Uβ from an old one, increasing the degree by exactly one in the process. Iterating this construction therefore gives many more examples of finite-degree twisted coefficient systems on Uβ.…”

A note on representations of welded braid groups

“…For instance, it reconstructs the unreduced Burau representation from a one dimensional representation. It was studied from a functorial point of view and extended in [24] and then generalised to other families of groups [23]. In particular, the underlying framework of this method, called the Long-Moody construction, naturally arises considering representations of welded braid groups: the aim of this section is the study of this construction in this case.…”

“…At the first step we obtain the Burau representation (Theorem 2.4) as in the case of B n . Surprisingly at the second iteration we do not obtain any new information, since we get the tensor product of two Burau representations (Theorem 2.6), while in the case of B n we recover this way Lawrence-Krammer representation (see [24,Section 2.3.1]). This result can be also compared with the fact that the "trivial" extension of Bigelow representation to wB n (associating to "braid" generators corresponding Bigelow matrices and to "permutation" generators the corresponding permutations matrices) is not well defined (see [3,14]).…”

“…The main idea of this work is to extend Long-Moody procedure [7,20,18,19,24,23] to wB n : in §1 we recall briefly the interpretation of welded braid groups in terms of fundamental groups of configuration spaces of circles and as automorphisms of free groups through Artin homomophism; this latter interpretation will be extended to other possible representations, extending Wada representations of the classical braid group B n . In §1.…”

Some computations of stable twisted homology for mapping class groups

“…[28, Proposition 3.8] Let M be a pre-braided strict monoidal category and α : UQ −→ M be a strong monoidal functor. Then, the precomposition by α provides a functor P ol strong n (UG, R-Mod) → P ol strong n (M, R-Mod).…”