Representation stability and finite linear groups

Abstract

We construct analogues of FI-modules where 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. Applications include a proof of the Lannes--Schwartz Artinian conjecture in the generic representation theory of finite fields, very general homological stability theorems with twisted coefficients for the general linear and symplectic groups over finite rings, and representation-theoretic versions of homological stability for congruence subgroups of the general linear group, the automorphism group of a free group, the symplectic group, and the mapping class group.

arXiv comment53 pages, 5 figures; major revision; to appear in Duke Math J

References in corpus (24)

• FI-modules and stability for representations of symmetric groups
• Representation theory and homological stability
• FI-modules over Noetherian rings
• Stabilization for mapping class groups of 3-manifolds
• Gröbner methods for representations of combinatorial categories
• GL-equivariant modules over polynomial rings in infinitely many variables
• Stability patterns in representation theory
• Homological stability for configuration spaces of manifolds
• Homology stability for outer automorphism groups of free groups
• Introduction to twisted commutative algebras
• An elementary illustrated introduction to simplicial sets
• Bounded-rank tensors are defined in bounded degree
• Noetherian property of infinite EI categories
• Representation Stability
• The Picard group of the moduli space of curves with level structures
• Stability in the homology of congruence subgroups
• Homological stability for automorphism groups
• Hilbert series for twisted commutative algebras
• FI_W-modules and stability criteria for representations of the classical Weyl groups
• Generating the Johnson filtration
• On finiteness properties of polynomial functors
• Generic representation theory of finite fields in nondescribing characteristic
• Le foncteur V -> $F_2[V{]}^{otimes3}$ entre F2-espaces vectoriels est noethérien
• Homology stability for unitary groups

Cited by in corpus (60)

• FI-modules and stability for representations of symmetric groups
• FI-modules over Noetherian rings
• Gröbner methods for representations of combinatorial categories
• FI-modules and the cohomology of modular representations of symmetric groups
• Representations of categories of G-maps
• FI- and OI-modules with varying coefficients
• Noetherian property of infinite EI categories
• Categories of FI type: a unified approach to generalizing representation stability and character polynomials
• Homological stability for automorphism groups
• Noetherianity of some degree two twisted skew-commutative algebras
• Codimension and Projective Dimension up to Symmetry
• Central stability for the homology of congruence subgroups and the second homology of Torelli groups
• On finite generation of the Johnson filtrations
• Homological Invariants of FI-modules and FI_G-modules
• Higher order representation stability and ordered configuration spaces of manifolds
• Twisted homological stability for configuration spaces
• Homological degrees of representations of categories with shift functors
• The representation theory of Brauer categories I: triangular categories
• On the degree-wise coherence of FI_G-modules
• $\mathrm{VI}$ modules in non-describing characteristic, Part I
• Stability in the high-dimensional cohomology of congruence subgroups
• Homological stability for classical groups
• Improved homological stability for certain general linear groups
• Koszulity of directed categories in representation stability theory
• Representation stability for filtrations of Torelli groups
• Coinduction functor in representation stability theory
• Central stability homology
• Categorifications of rational Hilbert series and characters of $F{S}^{op}$ modules
• Stability in the homology of unipotent groups
• Representation stability, secondary stability, and polynomial functors
• The categorical graph minor theorem
• Linear stable range for homology of congruence subgroups via FI-modules
• Topological actions of Temperley-Lieb algebras and Representation Stability
• Equivariant group presentations and the second homology group of the Torelli group
• GL-algebras in positive characteristic I: the exterior algebra
• Effective and Infinite-Rank Superrigidity in the Context of Representation Stability
• The artinian conjecture (following Djament, Putman, Sam, and Snowden)
• Bounds on homological invariants of VI-modules
• Structures in representation stability
• Asymptotic behavior of symmetric ideals: A brief survey
• Homological finiteness of functors on an additive category and applications
• Word Measures on $G{L}_N(q)$ and Free Group Algebras
• Representation stability and outer automorphism groups
• A Kruskal-Katona type result and applications
• Some generalizations of Schur functors
• Groupes d'extensions et foncteurs polynomiaux
• A Survey of Representation Stability Theory
• VIC-modules over noncommutative rings
• Representation Stability and Finite Orthogonal Groups
• Classical homological stability from the point of view of cells
• Polynomial conditions and homology of FI-modules
• Polynomial representations of the Witt Lie algebra
• Sur la noeth{é}rianit{é} locale des foncteurs polynomiaux
• A Tensor Restriction Theorem over Finite Fields
• Partial Torelli groups and homological stability
• Higher representation stability for ordered configuration spaces and twisted commutative factorization algebras
• Koszulity, supersolvability, and Stirling representations
• Asymptotic behaviors of representations of graded categories with inductive functors
• Sheaves of modules on atomic sites and discrete representations of topological groups
• Essential extensions, the nilpotent filtration and the Arone-Goodwillie tower