Generalized Long-Moody functors

TL;DR

This paper extends the Long-Moody construction to a broader class of groups, introducing functors that generate new representations and analyzing their impact on polynomial degrees and homological stability.

Contribution

It generalizes the Long-Moody functor framework to groups beyond braid groups, such as mapping class groups, and studies their effects on polynomial functors.

Findings

Long-Moody functors increase polynomial degrees by one.

New representations are constructed for various groups.

Homological stability properties are analyzed in this context.

Abstract

In this paper, we generalize the principle of the Long-Moody construction for representations of braid groups to other groups, such as mapping class groups of surfaces. Namely, we introduce endofunctors over a functor category that encodes representations of a family of groups. They are called Long-Moody functors and provide new representations. In this context, notions of polynomial functors are defined and play an important role in the study of homological stability. We prove that, under additional assumptions, a Long-Moody functor increases the very strong and weak polynomial degrees of functors by one.