GL-equivariant modules over polynomial rings in infinitely many variables

DOI10.1090/tran/6355zbMath1436.13012arXiv1206.2233OpenAlexW3101713268MaRDI QIDQ2796516

Steven V. Sam, Andrew Snowden

Publication date: 29 March 2016

Published in: Transactions of the American Mathematical Society (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1206.2233

Mathematics Subject Classification ID

Symmetric functions and generalizations (05E05) Combinatorial aspects of representation theory (05E10) Actions of groups on commutative rings; invariant theory (13A50) Representations of quivers and partially ordered sets (16G20) Structure, classification theorems for modules and ideals in commutative rings (13C05) Syzygies, resolutions, complexes and commutative rings (13D02)

Related Items (74)

Noetherianity of some degree two twisted commutative algebras ⋮ 𝑆𝑝-equivariant modules over polynomial rings in infinitely many variables ⋮ Equivariant Hilbert series in non-Noetherian polynomial rings ⋮ Representation stability for syzygies of line bundles on Segre-Veronese varieties ⋮ Hilbert series for twisted commutative algebras ⋮ Regularity of 𝐅𝐈-modules and local cohomology ⋮ Upper bounds of homological invariants of \({FI_G}\)-modules ⋮ Coinduction functor in representation stability theory ⋮ Linear and quadratic ranges in representation stability ⋮ Categories of dimension zero ⋮ Some generalizations of Schur functors ⋮ Invariant Chains in Algebra and Discrete Geometry ⋮ Homology of Littlewood complexes ⋮ Asymptotic behavior of representations of graded categories with inductive functors ⋮ Theorems of Carathéodory, Minkowski–Weyl, and Gordan up to Symmetry ⋮ Topological noetherianity for cubic polynomials ⋮ Kronecker comultiplication of stable characters and restriction from \(S_{ mn }\) to \(S_m \times S_n\) ⋮ The Representation Theory of the Increasing Monoid ⋮ Generalized representation stability and 𝐹𝐼_{𝑑}-modules ⋮ Categorifications of rational Hilbert series and characters of \(\boldsymbol{FS}^{\mathrm{op}}\) modules ⋮ The spectrum of a twisted commutative algebra ⋮ Some examples of simple generic 𝐹𝐼-modules in positive characteristic ⋮ A classification of injective FI ^m -modules ⋮ Supernatural analogues of Beilinson monads ⋮ Shift invariant algebras, Segre products and regular languages ⋮ Regularity bounds for twisted commutative algebras ⋮ MFO-RIMS tandem workshop: Symmetries on polynomial ideals and varieties. Abstracts from the MFO-RIMS tandem workshop held September 5--11, 2021 (hybrid meeting) ⋮ Koszulity of directed categories in representation stability theory ⋮ On computing the eventual behavior of an 𝐹𝐼-module over the rational numbers ⋮ Topological Noetherianity of polynomial functors. II: Base rings with Noetherian spectrum ⋮ Homological invariants of \(\mathrm{FI}\)-modules and \(\mathrm{FI}_{G}\)-modules ⋮ Plücker varieties and higher secants of Sato's Grassmannian ⋮ Representation theory and homological stability. ⋮ Betti tables of monomial ideals fixed by permutations of the variables ⋮ Phylogenetic complexity of the Kimura 3-parameter model ⋮ Classifying global dimensions of category algebras of some finite combinational categories ⋮ Noetherianity of some degree two twisted skew-commutative algebras ⋮ \(FI\)-modules over preadditive categories and torsion theories ⋮ GL-EQUIVARIANT MODULES OVER POLYNOMIAL RINGS IN INFINITELY MANY VARIABLES. II ⋮ FI-modules over Noetherian rings. ⋮ Deligne categories and representations of the infinite symmetric group ⋮ Topology of arrangements and representation stability. Abstracts from the workshop held January 14--20, 2018 ⋮ Tangential varieties of Segre-Veronese varieties ⋮ Representation stability for sequences of 0-Hecke modules ⋮ Lefschetz properties in algebra, geometry and combinatorics. Abstracts from the workshop held September 27 -- October 3, 2020 (hybrid meeting) ⋮ \(\mathrm{FI}^{m}\)-modules over Noetherian rings ⋮ An application of Nakayama functor in representation stability theory ⋮ Local cohomology and the multigraded regularity of \(\mathcal{FI}^m\)-modules ⋮ Representations of categories of \(G\)-maps ⋮ Ideals of bounded rank symmetric tensors are generated in bounded degree ⋮ Depth and the local cohomology of \(\mathcal{FI}_G\)-modules ⋮ Categories of FI type: a unified approach to generalizing representation stability and character polynomials ⋮ FI-sets with relations ⋮ VI-modules in nondescribing characteristic. I ⋮ Noetherian properties in representation theory ⋮ An application of the theory of FI-algebras to graph configuration spaces ⋮ Proof of Stembridge's conjecture on stability of Kronecker coefficients ⋮ Relations between the \(2 \times 2\) minors of a generic matrix ⋮ \(\mathrm{FI}_{\mathcal W}\)-modules and stability criteria for representations of classical Weyl groups. ⋮ Rationality of equivariant Hilbert series and asymptotic properties ⋮ VI-modules in non-describing characteristic. II ⋮ Topological Noetherianity of polynomial functors ⋮ Gröbner methods for representations of combinatorial categories ⋮ FI- and OI-modules with varying coefficients ⋮ Representation stability and outer automorphism groups ⋮ On the local cohomology of L-shaped integral FI-modules ⋮ Configuration spaces of graphs with certain permitted collisions ⋮ Littlewood complexes for symmetric groups ⋮ Homological degrees of representations of categories with shift functors ⋮ Combinatorial categorical equivalences of Dold-Kan type ⋮ Tensor Representations of Mackey Lie Algebras and Their Dense Subalgebras ⋮ Resolutions of ideals of subspace arrangements ⋮ FI-modules and stability for representations of symmetric groups ⋮ Homology of FI-modules

Uses Software

• symmChainGens
• FourTiTwo
• PieriMaps

Cites Work

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• Noetherianity of some degree two twisted commutative algebras
• Pieri resolutions for classical groups.
• Finite Gröbner bases in infinite dimensional polynomial rings and applications
• The existence of equivariant pure free resolutions
• Finiteness for the \(k\)-factor model and chirality varieties
• Young diagrams and determinantal varieties
• Finiteness theorems and algorithms for permutation invariant chains of Laurent lattice ideals
• Syzygies of Segre embeddings and \(\Delta\)-modules
• Computing inclusions of Schur modules
• FI-modules and stability for representations of symmetric groups
• Character polynomials, their \(q\)-analogs and the Kronecker product.
• Bounded-rank tensors are defined in bounded degree
• Quivers and the cohomology of homogeneous vector bundles
• Gröbner methods for representations of combinatorial categories
• Local Cohomology Modules for Normal Domains
• Sheaf cohomology and free resolutions over exterior algebras
• The Geometry of Syzygies
• Des catégories abéliennes

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