Gröbner methods for representations of combinatorial categories

Abstract

Given a category C \mathcal {C} of a combinatorial nature, we study the following fundamental question: how do combinatorial properties of C \mathcal {C} affect algebraic properties of representations of C \mathcal {C} ? We prove two general results. The first gives a criterion for representations of C \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 C \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 Δ \Delta -modules, introduced by Snowden in connection to syzygies of Segre embeddings; and we establish fundamental properties of twisted commutative algebras in positive characteristic.

On central stability

“…It was proved by Sam and Snowden that for the categories in (i)‐(iii) above, every finitely generated graded ‐module over a commutative noetherian ring is noetherian (and hence they are centrally stable by Corollary ).…”

“…In particular, our proof does not require consideration of the complex or similar complexes. Moreover, the setting for our generalization is much simpler than the one of complemented categories with a generator used by Putman and Sam , and include many more examples; our generalization is, in fact, motivated by the fact that several combinatorial categories studied by Sam and Snowden in do not fall within the framework of . Another goal of our paper is to explain the role of the quadratic property of in the isomorphism .…”

“…We prove that the converse of the preceding statement holds under a local finiteness assumption on the category when the base ring is a commutative noetherian ring. To illustrate the wide applicability of our results, we recall examples of combinatorial categories introduced by Sam and Snowden in which fall within our framework but are not complemented categories with a generator.…”

An application of the theory of $${{\,\mathrm{FI}\,}}$$FI-algebras to graph configuration spaces

“…The representation theory of FI has recently risen to popularity due in large part to the seminal works of authors such as Church, Ellenberg, Farb, and Nagpal [CEF, CEFN], Djament and Vespa [D, DV], Sam and Snowden [SS,SS2], and many others. In this work we will be primarily concerned with two constructions stemming from FI: FI-algebras and FI-graphs.…”

Essential extensions, the nilpotent filtration and the Arone–Goodwillie tower

“…It is now known that (at least up to nilpotent unstable modules) there are few restrictions which can be placed on ρ n M (for n 1), following the affirmation of the Lannes and Schwartz Artinian conjecture [19,20]. Namely, generalizing Kuhn's observation [11], examples can be manufactured by considering the topological realization of the beginning of an injective resolution (modulo nilpotents) of a given reduced module and forming the homotopy cofibre:…”