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.

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

References in corpus (8)

• Gröbner methods for representations of combinatorial categories
• Introduction to twisted commutative algebras
• Noetherian property of infinite EI categories
• Representation Stability
• Homological stability for automorphism groups
• Hilbert series for twisted commutative algebras
• Generic representation theory of finite fields in nondescribing characteristic
• Le foncteur V -> $F_2[V{]}^{otimes3}$ entre F2-espaces vectoriels est noethérien

Cited by in corpus (57)

• 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
• 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
• Coinduction functor in representation stability theory
• Representation stability for filtrations of Torelli groups
• Central stability homology
• Representation stability, secondary stability, and polynomial functors
• Stability in the homology of unipotent groups
• Categorifications of rational Hilbert series and characters of $F{S}^{op}$ modules
• Linear stable range for homology of congruence subgroups via FI-modules
• The categorical graph minor theorem
• Topological actions of Temperley-Lieb algebras and Representation Stability
• The artinian conjecture (following Djament, Putman, Sam, and Snowden)
• 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
• Representation stability and outer automorphism groups
• A Kruskal-Katona type result and applications
• Bounds on homological invariants of VI-modules
• Word Measures on $G{L}_N(q)$ and Free Group Algebras
• Some generalizations of Schur functors
• Asymptotic behavior of symmetric ideals: A brief survey
• Structures in representation stability
• Homological finiteness of functors on an additive category and applications
• VIC-modules over noncommutative rings
• Representation Stability and Finite Orthogonal Groups
• Groupes d'extensions et foncteurs polynomiaux
• Classical homological stability from the point of view of cells
• Polynomial conditions and homology of FI-modules
• Polynomial representations of the Witt Lie algebra
• A Survey of Representation Stability Theory
• A Tensor Restriction Theorem over Finite Fields
• Koszulity, supersolvability, and Stirling representations
• Essential extensions, the nilpotent filtration and the Arone-Goodwillie tower
• Sur la noeth{é}rianit{é} locale des foncteurs polynomiaux
• Asymptotic behaviors of representations of graded categories with inductive functors
• Partial Torelli groups and homological stability
• Sheaves of modules on atomic sites and discrete representations of topological groups
• Higher representation stability for ordered configuration spaces and twisted commutative factorization algebras