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Generalized representation stability and $ \mathrm{FI}_d$-modules


Author: Eric Ramos
Journal: Proc. Amer. Math. Soc. 145 (2017), 4647-4660
MSC (2010): Primary 05E10, 16G20, 18A25; Secondary 13D15
DOI: https://doi.org/10.1090/proc/13618
Published electronically: June 9, 2017
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Abstract: In this note we consider the complex representation theory of $ \mathrm {FI}_d$, a natural generalization of the category $ \mathrm {FI}$ of finite sets and injections. We prove that finitely generated $ \mathrm {FI}_d$-modules exhibit behaviors in the spirit of Church-Farb representation stability theory, generalizing a theorem of Church, Ellenberg, and Farb which connects finite generation of $ \mathrm {FI}$-modules to representation stability.


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Additional Information

Eric Ramos
Affiliation: Department of Mathematics, University of Wisconsin–Madison, Madison, Wisconsin 53706
Email: eramos@math.wisc.edu

DOI: https://doi.org/10.1090/proc/13618
Keywords: Representation stability, $\mathrm{FI}$-modules
Received by editor(s): November 11, 2016
Received by editor(s) in revised form: December 5, 2016
Published electronically: June 9, 2017
Additional Notes: The author was supported by NSF grant DMS-1502553
Communicated by: Jerzy Weyman
Article copyright: © Copyright 2017 American Mathematical Society