Gröbner methods for representations of combinatorial categories

By Steven V. Sam, Andrew Snowden. 2014

212 citations recorded by the source corpus.

Given a category C of a combinatorial nature, we study the following fundamental question: how does the combinatorial behavior of C affect the algebraic behavior of representations of C? We prove two general results. The first gives a combinatorial criterion for representations of C to admit a theory of Gr\"obner bases. From this, we obtain a criterion for noetherianity of representations. The second gives a combinatorial criterion for a general "rationality" result for Hilbert series of representations of C. This criterion connects to the theory of formal languages, and makes essential use of results on the generating functions of languages, such as the transfer-matrix method and the Chomsky-Sch\"utzenberger theorem. 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 (originally introduced by Church-Ellenberg-Farb), and a family of natural generalizations, are noetherian; we give an easy proof of a generalization of 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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d83e2ffe25d844ff1fde4051a2aa4625433f597e

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Citations in the corpus

Representation stability and finite linear groups (2014, methodology): “ The total ordering that we deﬁne for the proof of Lemma 2.10 need not be compatible with all compositions in the sense that f′ <f implies that gf′ <gf for all g. This assumption is in place in [SamSn4] which is why we cannot use the machinery from that paper. To see this, consider the example”