GL-equivariant modules over polynomial rings in infinitely many variables

Abstract

Consider the polynomial ring in countably infinitely many variables over a field of characteristic zero, together with its natural action of the infinite general linear group G. We study the algebraic and homological properties of finitely generated modules over this ring that are equipped with a compatible G-action. We define and prove finiteness properties for analogues of Hilbert series, systems of parameters, depth, local cohomology, Koszul duality, and regularity. We also show that this category is built out of a simpler, more combinatorial, quiver category which we describe explicitly. Our work is motivated by recent papers in the literature which study finiteness properties of infinite polynomial rings equipped with group actions. (For example, the paper by Church, Ellenberg and Farb on the category of FI-modules, which is equivalent to our category.) Along the way, we see several connections with the character polynomials from the representation theory of the symmetric groups. Several examples are given to illustrate that the invariants we introduce are explicit and computable.

59 pages, uses ytableau.sty; v2: expanded details in many proofs especially in Sections 2 and 4, Section 6 substantially expanded, added references; v3: corrected typos and Remark 4.3.3 from published version

References in corpus (2)

• Stability patterns in representation theory
• Introduction to twisted commutative algebras

Cited by in corpus (56)

• FI-modules over Noetherian rings
• Gröbner methods for representations of combinatorial categories
• Stability patterns in representation theory
• Representation stability and finite linear groups
• Homology of FI-modules
• Introduction to twisted commutative algebras
• FI-modules and the cohomology of modular representations of symmetric groups
• Equivariant Hilbert Series in non-Noetherian Polynomial Rings
• Representations of categories of G-maps
• Noetherianity of some degree two twisted commutative algebras
• FI- and OI-modules with varying coefficients
• Noetherian property of infinite EI categories
• Castelnuovo-Mumford regularity up to symmetry
• Categories of FI type: a unified approach to generalizing representation stability and character polynomials
• Proof of Stembridge's conjecture on stability of Kronecker coefficients
• Noetherianity of some degree two twisted skew-commutative algebras
• GL-equivariant modules over polynomial rings in infinitely many variables. II
• A long exact sequence for homology of FI-modules
• Topological noetherianity for cubic polynomials
• Periodicity in the cohomology of symmetric groups via divided powers
• Regularity of FI-modules and local cohomology
• Syzygies of bounded rank symmetric tensors are generated in bounded degree
• Ideals of bounded rank symmetric tensors are generated in bounded degree
• The representation theory of Brauer categories I: triangular categories
• Hilbert series for twisted commutative algebras
• FI_W-modules and stability criteria for representations of the classical Weyl groups
• Representation stability for syzygies of line bundles on Segre--Veronese varieties
• $\mathrm{VI}$ modules in non-describing characteristic, Part I
• Combinatorial categorical equivalences of Dold-Kan type
• Uniformly Presented Vector Spaces
• Coinduction functor in representation stability theory
• Sp-equivariant modules over polynomial rings in infinitely many variables
• Invariant chains in algebra and discrete geometry
• Categorifications of rational Hilbert series and characters of $F{S}^{op}$ modules
• Relations between the 2x2 minors of a generic matrix
• Finite phylogenetic complexity and combinatorics of tables
• Topological Noetherianity of polynomial functors II: base rings with Noetherian spectrum
• Theorems of Carathéodory, Minkowski-Weyl, and Gordan up to symmetry
• Regularity bounds for twisted commutative algebras
• Supernatural analogues of Beilinson monads
• GL-algebras in positive characteristic I: the exterior algebra
• Some generalizations of Schur functors
• Stable representation theory: beyond the classical groups
• Structures in representation stability
• Representation stability for sequences of 0-Hecke modules
• Representation stability and outer automorphism groups
• The semi-linear representation theory of the infinite symmetric group
• Representation theory of Mackey Lie algebras and their dense subalgebras
• Elementary Equivariant Modules
• GL-algebras in positive characteristic II: the polynomial ring
• On the tails of FI-modules
• Stillman's question for twisted commutative algebras
• Sheaves of modules on atomic sites and discrete representations of topological groups
• A note on projective dimension over twisted commutative algebras
• Kronecker Comultiplication of Stable Characters and Restriction From $S_{mn}$ to $S_m\times S_n$
• A counterexample to a Gröbner approach for noetherianity of the twisted commutative algebra $\mathrm{Sym}({\mathrm{Sym}}^2({\mathbf{C}}^{\infty }))$