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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Gröbner methods for representations of combinatorial categories
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by Steven V Sam and Andrew Snowden PDF
J. Amer. Math. Soc. 30 (2017), 159-203 Request permission

Abstract:

Given a category $\mathcal {C}$ of a combinatorial nature, we study the following fundamental question: how do combinatorial properties of $\mathcal {C}$ affect algebraic properties of representations of $\mathcal {C}$? We prove two general results. The first gives a criterion for representations of $\mathcal {C}$ to admit a theory of Gröbner bases, from which we obtain a criterion for noetherianity. The second gives a criterion for a general “rationality” result for Hilbert series of representations of $\mathcal {C}$, and connects to the theory of formal languages.

Our work is motivated by recent work in the literature on representations of various specific categories. Our general criteria recover many of the results on these categories that had been proved by ad hoc means, and often yield cleaner proofs and stronger statements. For example, we give a new, more robust, proof that FI-modules (studied by Church, Ellenberg, and Farb), and certain generalizations, are noetherian; we prove the Lannes–Schwartz artinian conjecture from the study of generic representation theory of finite fields; we significantly improve the theory of $\Delta$-modules, introduced by Snowden in connection to syzygies of Segre embeddings; and we establish fundamental properties of twisted commutative algebras in positive characteristic.

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Additional Information
  • Steven V Sam
  • Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
  • Address at time of publication: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
  • MR Author ID: 836995
  • ORCID: 0000-0003-1940-9570
  • Email: svs@math.wisc.edu
  • Andrew Snowden
  • Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
  • MR Author ID: 788741
  • Email: asnowden@umich.edu
  • Received by editor(s): September 21, 2014
  • Received by editor(s) in revised form: June 21, 2015, September 4, 2015, and February 4, 2016
  • Published electronically: March 17, 2016
  • Additional Notes: The first author was supported by a Miller research fellowship.
    The second author was supported by NSF grant DMS-1303082.
  • © Copyright 2016 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 30 (2017), 159-203
  • MSC (2010): Primary 05A15, 13P10, 16P40, 18A25, 68Q70
  • DOI: https://doi.org/10.1090/jams/859
  • MathSciNet review: 3556290