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Gröbner methods for representations of combinatorial categories


Authors: Steven V Sam and Andrew Snowden
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
Published electronically: March 17, 2016
MathSciNet review: 3556290
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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
Email: svs@math.wisc.edu

Andrew Snowden
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: asnowden@umich.edu

DOI: https://doi.org/10.1090/jams/859
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.
Article copyright: © Copyright 2016 American Mathematical Society