An inductive machinery for representations of categories with shift functors

2016/10/28 by Wee Liang Gan, Liping Li, Gan, Wee Liang +1 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1610.09081

openalex publication_date 2016/10/28 · openalex created_date 2016/11/04 · openalex updated_date 2026/07/28

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 \FI_G and \OI_G which appear in representation stability theory.

Citations

• Depth and the Local Cohomology of FI_G-modules

Cited by

• On representations of permutation groups and orbit categories

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• Asymptotic behaviors of representations of graded categories with inductive functors
• Castelnuovo-Mumford regularity of representations of certain product categories
• Homological degrees of representations of categories with shift functors
• Extensions of representation stable categories
• Invertibility in Category Representations