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Generalized Long–Moody functors

Arthur Soulié

Algebraic & Geometric Topology 22 (2022) 1713–1788
Abstract

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.

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Keywords
Long–Moody construction, functor categories, mapping class groups, polynomial functors
Mathematical Subject Classification 2010
Primary: 20C07, 20F36, 20F38, 20J99, 57M07
Secondary: 18A25, 18D10, 57N05
References
Publication
Received: 11 December 2019
Revised: 2 June 2021
Accepted: 16 June 2021
Published: 10 October 2022
Authors
Arthur Soulié
School of Mathematics and Statistics
University of Glasgow
Glasgow
United Kingdom
Center for Geometry and Physics
Institute for Basic Science and POSTECH
Pohang
South Korea