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• W3102979608 abstract "Given a category <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper C> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>C</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a combinatorial nature, we study the following fundamental question: how do combinatorial properties of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper C> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>C</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> affect algebraic properties of representations of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper C> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>C</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>? We prove two general results. The first gives a criterion for representations of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper C> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>C</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper C> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>C</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, 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 <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Delta> <mml:semantics> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> <mml:annotation encoding=application/x-tex>Delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-modules, introduced by Snowden in connection to syzygies of Segre embeddings; and we establish fundamental properties of twisted commutative algebras in positive characteristic." @default.
• W3102979608 created "2020-11-23" @default.
• W3102979608 creator A5009159816 @default.
• W3102979608 creator A5066686947 @default.
• W3102979608 date "2016-03-17" @default.
• W3102979608 modified "2023-10-18" @default.
• W3102979608 title "Gröbner methods for representations of combinatorial categories" @default.
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• W3102979608 doi "https://doi.org/10.1090/jams/859" @default.
• W3102979608 hasPublicationYear "2016" @default.
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