Gröbner methods for representations of combinatorial categories

Sam, Steven; Snowden, Andrew

doi:10.1090/jams/859

Gröbner methods for representations of combinatorial categories
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by Steven V Sam and Andrew Snowden;

J. Amer. Math. Soc. 30 (2017), 159-203

DOI: https://doi.org/10.1090/jams/859

Published electronically: March 17, 2016

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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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