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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



An inductive machinery for representations of categories with shift functors

Authors: Wee Liang Gan and Liping Li
Journal: Trans. Amer. Math. Soc. 371 (2019), 8513-8534
MSC (2010): Primary 16E30, 16G99, 16P40
Published electronically: February 22, 2019
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Abstract: We describe an inductive machinery to prove various properties of representations of a category equipped with a generic shift functor. Specifically, we show that if a property (P) of representations of the category behaves well under the generic shift functor, then all finitely generated representations of the category have the property (P). In this way, we obtain simple criteria for properties such as Noetherianity, finiteness of Castelnuovo-Mumford regularity, and polynomial growth of dimension to hold. This gives a systemetic and uniform proof of such properties for representations of the categories $ \mathscr {FI}_G$ and $ \mathscr {OI}_G$ which appear in representation stability theory.

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Additional Information

Wee Liang Gan
Affiliation: Department of Mathematics, University of California, Riverside, California 92521

Liping Li
Affiliation: LCSM (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, China

Received by editor(s): February 21, 2017
Received by editor(s) in revised form: April 21, 2017, and October 19, 2017
Published electronically: February 22, 2019
Additional Notes: The second author was supported by the National Natural Science Foundation of China 11771135, the Construct Program of the Key Discipline in Hunan Province, and the Start-Up Funds of Hunan Normal University 830122-0037.
Article copyright: © Copyright 2019 American Mathematical Society