Gröbner methods for representations of combinatorial categories

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 Gröbner 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-Schützenberger 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.

41 pages; v2: Moved old Sections 3.4, 10, 11, 13.2 and connected text to arxiv:1410.6054v1, Section 13.1 removed and will appear elsewhere; v3: substantial revision and reorganization of sections

References in corpus (5)

• Representation stability and finite linear groups
• Introduction to twisted commutative algebras
• Noetherianity of some degree two twisted commutative algebras
• Pattern avoidance is not P-recursive
• The artinian conjecture (following Djament, Putman, Sam, and Snowden)

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• Periodicity in the cohomology of symmetric groups via divided powers
• Regularity of FI-modules and local cohomology
• The representation theory of Brauer categories I: triangular categories
• Syzygies of bounded rank symmetric tensors are generated in bounded degree
• Ideals of bounded rank symmetric tensors are generated in bounded degree
• Hilbert series for twisted commutative algebras
• $\mathrm{VI}$ modules in non-describing characteristic, Part I
• Invariant chains in algebra and discrete geometry
• Stability in the homology of unipotent groups
• Configuration spaces of disks in a strip, twisted algebras, persistence, and other stories
• Categorifications of rational Hilbert series and characters of $F{S}^{op}$ modules
• A generating function approach to new representation stability phenomena in orbit configuration spaces
• Theorems of Carathéodory, Minkowski-Weyl, and Gordan up to symmetry
• Regularity bounds for twisted commutative algebras
• GL-algebras in positive characteristic I: the exterior algebra
• A Kruskal-Katona type result and applications
• Homological finiteness of functors on an additive category and applications
• Asymptotic growth of Betti numbers of ordered configuration spaces on an elliptic curve
• Asymptotic behavior of symmetric ideals: A brief survey
• Representation stability for sequences of 0-Hecke modules
• Representation Stability and Finite Orthogonal Groups
• VIC-modules over noncommutative rings
• Polynomial representations of the Witt Lie algebra
• On finite generation in magnitude (co)homology, and its torsion
• Essential extensions, the nilpotent filtration and the Arone-Goodwillie tower
• Sur la noeth{é}rianit{é} locale des foncteurs polynomiaux
• A Tensor Restriction Theorem over Finite Fields
• On monoids up to symmetry
• Higher representation stability for ordered configuration spaces and twisted commutative factorization algebras
• Periodicity in the cohomology of finite general linear groups via q-divided powers