Representation theory and homological stability

Abstract

We introduce the idea of *representation stability* (and several variations) for a sequence of representations V_n of groups G_n. A central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability. This makes it possible to extend classical theorems of homological stability to a much broader variety of examples. Representation stability also provides a framework in which to find and to predict patterns, from classical representation theory (Littlewood--Richardson and Murnaghan rules, stability of Schur functors), to cohomology of groups (pure braid, Torelli and congruence groups), to Lie algebras and their homology, to the (equivariant) cohomology of flag and Schubert varieties, to combinatorics (the (n+1)^(n-1) conjecture). The majority of this paper is devoted to exposing this phenomenon through examples. In doing this we obtain applications, theorems and conjectures. Beyond the discovery of new phenomena, the viewpoint of representation stability can be useful in solving problems outside the theory. In addition to the applications given in this paper, it is applied in [CEF] to counting problems in number theory and finite group theory. Representation stability is also used in [C] to give broad generalizations and new proofs of classical homological stability theorems for configuration spaces on oriented manifolds.

91 pages. v2: minor revisions throughout. v3: final version, to appear in Advances in Mathematics

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• FI-modules and stability for representations of symmetric groups
• FI-modules over Noetherian rings
• Gröbner methods for representations of combinatorial categories
• Stability patterns in representation theory
• Representation stability and finite linear groups
• Homological stability for configuration spaces of manifolds
• FI-modules and the cohomology of modular representations of symmetric groups
• Homological stability for unordered configuration spaces
• Representation stability in cohomology and asymptotics for families of varieties over finite fields
• Topology of moduli spaces of tropical curves with marked points
• A spectral sequence for stratified spaces and configuration spaces of points
• Assembling homology classes in automorphism groups of free groups
• Representation Stability
• Categories of FI type: a unified approach to generalizing representation stability and character polynomials
• Stability in the homology of congruence subgroups
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• The homology of the Higman-Thompson groups
• The tropicalization of the moduli space of curves II: Topology and applications
• GL-equivariant modules over polynomial rings in infinitely many variables. II
• Central stability for the homology of congruence subgroups and the second homology of Torelli groups
• Kazhdan-Lusztig polynomials of matroids: a survey of results and conjectures
• Higher order representation stability and ordered configuration spaces of manifolds
• Homological Invariants of FI-modules and FI_G-modules
• Word Measures on Symmetric Groups
• Symmetric group characters as symmetric functions
• Representation stability for cohomology of configuration spaces in ${\mathbf{R}}^d$
• Homological degrees of representations of categories with shift functors
• Depth and the Local Cohomology of FI_G-modules
• Twisted homological stability for configuration spaces
• Homological stability of topological moduli spaces
• Representation stability for syzygies of line bundles on Segre--Veronese varieties
• $\mathrm{VI}$ modules in non-describing characteristic, Part I
• The homology of the Brauer algebras
• Stability and periodicity in the modular representation theory of symmetric groups
• FI_W-modules and stability criteria for representations of the classical Weyl groups
• On the second homology group of the Torelli subgroup of Aut(F_n)
• Filtrations and Homological degrees of FI-modules
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• Equivariant Topology of Configuration Spaces
• Algebraic structures on cohomology of configuration spaces of manifolds with flows
• Dwyer-Kan homotopy theory of enriched categories
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• Uniformly Presented Vector Spaces
• Homological stability for spaces of commuting elements in Lie groups
• Representation stability for filtrations of Torelli groups
• Coinduction functor in representation stability theory
• Twisted cohomology of configuration spaces and spaces of maximal tori via point-counting
• Homology of braid groups, the Burau representation, and ${\mathbb{F}}_q$-points on superelliptic curves
• Spectra of Symmetrized Shuffling Operators
• On the cohomology of pure mapping class groups as FI-modules
• Central stability homology
• FI_W-modules and constraints on classical Weyl group characters
• Twisted homological stability for extensions and automorphism groups of free nilpotent groups
• Automorphisms of free groups, I
• ${\mathrm{FI}}_G$-modules, orbit configuration spaces, and complex reflection groups
• Polynomial splitting measures and cohomology of the pure braid group
• Lattice Spectral Sequences and Cohomology of Configuration Spaces
• Categorifications of rational Hilbert series and characters of $F{S}^{op}$ modules
• Splitting Behavior of $S_n$-Polynomials
• An application of the theory of FI-algebras to graph configuration spaces
• Representation stability, secondary stability, and polynomial functors
• Homological stability for moduli spaces of disconnected submanifolds, I
• The equivariant Kazhdan-Lusztig polynomial of a matroid
• Representation theory for the Kriz model
• Motivic random variables and representation stability I: Configuration spaces
• A comparison of twisted coefficient systems
• Equivariant group presentations and the second homology group of the Torelli group
• Graph complexes and the symplectic character of the Torelli group
• On the growth of the Kronecker coefficients
• Generating functions and statistics on spaces of maximal tori in classical Lie groups
• Representation stability of power sets and square free polynomials
• Simplicial $G$-complexes and representation stability of polyhedral products
• Polynomial factorization statistics and point configurations in ${\mathbb{R}}^3$
• Symmetric group characters as symmetric functions (extended abstract)
• Asymptotic growth of Betti numbers of ordered configuration spaces on an elliptic curve
• Representation stability and outer automorphism groups
• Hecke Operators on Stable Cohomology
• VIC-modules over noncommutative rings
• Generalized representation stability for disks in a strip and no-k-equal spaces
• Representation Stability and Finite Orthogonal Groups
• Regularity and stable ranges of FI-modules
• On computing the eventual behavior of an FI-module over the rational numbers
• A Survey of Representation Stability Theory
• Generalized Representation Stability and FI_d-modules
• Local cohomology and the multi-graded regularity of FI${}^m$-modules
• Stability of Springer representations for type $A$
• Homology of the braid group with coefficients in the reduced Burau representation
• On monoids up to symmetry
• Asymptotic behaviors of representations of graded categories with inductive functors
• On the equivalence of two stability conditions of FB-modules
• The symmetric group action on rank-selected posets of injective words
• Partial Torelli groups and homological stability
• An inductive method for $\mathrm{OI}$-modules
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• Families of cellular resolutions, their syzygies, and stability
• Secondary characteristic classes for subgroups of automorphism groups of free groups
• Higher representation stability for ordered configuration spaces and twisted commutative factorization algebras
• Representation stability for homotopy automorphisms
• The ${\mathbb{S}}_n$-equivariant Euler characteristic of the moduli space of graphs
• Stable homology of Lie algebras of derivations and homotopy invariants of wheeled operads
• Koszulity, supersolvability, and Stirling representations
• Symmetric Ideals and Invariant Hilbert Schemes
• A geometric approach to counting norms in cyclic extensions of function fields
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• Generators for the representation rings of certain wreath products
• On the Homology of Nilpotent k-ary Lie Algebras
• Products of stabilizing representations
• An efficient algorithm computing composition factors of $T(V{)}^{\otimes n}$
• Decomposing induced characters of the centralizer of an $n$-cycle in the symmetric group on $2n$-elements
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• Factorization statistics and the twisted Grothendieck-Lefschetz formula