Gröbner methods for representations of combinatorial categories
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In this article, the authors studied how the combinatorial behavior of a category C affects the algebraic behavior of representations of C, and showed that C-algebraic representations are noetherian.Abstract:
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 Grobner 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–Schutzenberger 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 ∆modules, introduced by Snowden in connection to syzygies of Segre embeddings; and we establish fundamental properties of twisted commutative algebras in positive characteristic.read more
Citations
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Journal ArticleDOI
Representation stability and finite linear groups
Andrew Putman,Steven V Sam +1 more
TL;DR: In this paper, the role of the symmetric group is played by the general linear groups and the symplectic groups over finite rings and prove basic structural properties such as Noetherianity.
Journal ArticleDOI
Representation stability and finite linear groups
Andrew Putman,Steven V Sam +1 more
TL;DR: In this paper, the role of the symmetric group is played by the general linear groups and the symplectic groups over finite rings, and basic structural properties such as local Noetherianity are proved.
Journal ArticleDOI
Equivariant Hilbert series in non-noetherian polynomial rings
Uwe Nagel,Tim Römer +1 more
TL;DR: In this article, the authors studied equivariant Hilbert series of ideals in polynomial rings in countably many variables that are invariant under a suitable action of a symmetric group or the monoid Inc ( N ) of strictly increasing functions.
Journal ArticleDOI
Topological Noetherianity of polynomial functors
TL;DR: In this paper, it was shown that any finite-degree polynomial functor over an infinite field is topologically Noetherian, which implies boundedness of a wider class of invariants of ideals.
Posted Content
Noetherian property of infinite EI categories
Wee Liang Gan,Liping Li +1 more
TL;DR: In this article, the authors generalize this result to the abstract setting of an infinite EI category satisfying certain combinatorial conditions and show that finitely generated FI-modules over a field of characteristic 0 are Noetherian.
References
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Book
Introduction to Automata Theory, Languages, and Computation
TL;DR: This book is a rigorous exposition of formal languages and models of computation, with an introduction to computational complexity, appropriate for upper-level computer science undergraduates who are comfortable with mathematical arguments.
Book
Symmetric functions and Hall polynomials
TL;DR: In this paper, the characters of GLn over a finite field and the Hecke ring of GLs over finite fields have been investigated and shown to be symmetric functions with two parameters.
Book
Graduate Texts in Mathematics
Rajendra Bhatia,Glen Bredon,Wolfgang Walter,Joseph J. Rotman,M. Ram Murty,Jane Gilman,Peter Walters,Martin Golubitsky,Ioannis Karatzas,Henri Cohen,Raoul Bott,Gaisi Takeuti,Béla Bollobás,John M. Lee,Jiří Matoušek,Saunders Mac Lane,John L. Kelley,B. A. Dubrovin,Tom M. Apostol,John Stillwell,William Arveson +20 more
Book
Enumerative Combinatorics: Volume 1
TL;DR: The second edition of the Basic Introduction to Enumerative Combinative Analysis as discussed by the authors includes ten new sections and more than 300 new exercises, most with solutions, reflecting numerous new developments since the publication of the first edition in 1986.
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