Representation theory and homological stability.
type: Item
instance of: scholarly article
title: Representation theory and homological stability.
zbMATH Open document ID: 1300.20051
author: Thomas Church
author: Benson Farb
published in: Advances in Mathematics
publication date: 2014-03-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/1008.1368
review text: The authors analyze sequences of representations, \(\{V_n\}\), of families of groups, \(\{G_n\}\). The groups are either the special linear/symplectic families, or their Weyl groups. The starting point is the well-known fact that these families have a built-in stability, concerning the names for the irreducible representations of the groups in the family. The authors define (uniform) representation stability for \(\{V_n\}\), by requiring the multiplicity of every irreducible label in \(V_n\) to be independent of \(n\) (periodic in \(n\), in the modular case), for \(n\) big enough. When the sequence \(\{V_n\}\) comes endowed with equivariant maps relating \(V_n\) to \(V_{n+1}\), they also define refined versions of stability, which also take into account the behaviour of these maps. They go on proving that stability is preserved by natural operations, such as tensor product and (iterated) Schur functors. Representation stability is established in an important example: cohomology of pure braid groups associated to the Coxeter arrangements of hyperplanes of type \(A\) and \(B\) (for which usual homological stability fails). The importation of representation theory into the homological study of configuration spaces enabled the first author to obtain broad generalizations of classical homological stability theorems, for configuration spaces of arbitrary oriented manifolds [Invent. Math. 188, No. 2, 465-504 (2012; Zbl 1244.55012)]. Another representation stability result, for the cohomology of associated flag manifolds, is proved in the paper under review, and related in subsequent work by the authors and Ellenberg to arithmetic statistics. The paper under review also contains an inciting conjectural picture, formulated in terms of representation stability, for the homology of Torelli groups (associated to either compact oriented surfaces with one boundary component, or free groups). In the modular case: it is shown that the torsion of the Abelianization of the above Tore…
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 19B14
Mathematics Subject Classification ID: 55R80
zbMATH DE Number: 6264399
zbMATH Keywords: irreducible representations
zbMATH Keywords: uniform representation stability
zbMATH Keywords: homological stability
zbMATH Keywords: braid groups
reviewed by: Ştefan Papadima
MaRDI profile type: Publication
OpenAlex ID: W2037343549
arXiv ID: 1008.1368
cites work: Automorphisms and cohomology of discrete groups
cites work: The cohomology ring of the colored braid group
cites work: Galois representations attached to mod \(p\) cohomology of \(GL(n,\mathbb{Z})\)
cites work: Representation Stability of Power Sets and Square Free Polynomials
cites work: Q3996716
cites work: Solution of the congruence subgroup problem for \(\text{SL}_ n\) \((n\geq 3)\) and \(\text{Sp}_{2n}\) \((n\geq 2)\)
cites work: Dimension of the Torelli group for \(\text{Out}(F_n)\).
cites work: The dimension of the Torelli group
cites work: Towards representation stability for the second homology of the Torelli group.
cites work: Density properties for certain subgroups of semi-simple groups without compact components
cites work: Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts
cites work: The stability of the Kronecker product of Schur functions
cites work: Q4403463
cites work: Q5752746
cites work: The Adjoint Homology of the Free 2-Step Nilpotent Lie Algebra
cites work: The cohomology of the cotangent bundle of Heisenberg groups.
cites work: Completed cohomology -- a survey
cites work: On the problem of homology stability for congruence subgroups
cites work: Homological stability for configuration spaces of manifolds
cites work: Representation stability in cohomology and asymptotics for families of varieties over finite fields
cites work: FI-modules and stability for representations of symmetric groups
cites work: FI-modules over Noetherian rings.
cites work: Parameterized Abel-Jacobi maps and abelian cycles in the Torelli group
cites work: Stability phenomena in the topology of moduli spaces
cites work: On the cohomology of an arrangement of type \(B_ l\)
cites work: The Minimal Model of the Complement of an Arrangement of Hyperplanes
cites work: Q3129746
cites work: Q4002278
cites work: Stable homology of automorphism groups of free groups.
cites work: Q4509026
cites work: Q4464601
cites work: Infinitesimal presentations of the Torelli groups
cites work: Q4011175
cites work: Torelli Groups and Geometry of Moduli Spaces of Curves
cites work: On the Complete GL(n, C)-Decomposition of the Stable Cohomology of gl n (A)
cites work: Stable decompositions for some symmetric group characters arising in braid group cohomology.
cites work: Q4750828
cites work: Q5715118
cites work: The integral cohomology of the group of loops.
cites work: Representation stability for the cohomology of the moduli space \(\mathcal M_{g}^{n}\)
cites work: On the cohomology of pure mapping class groups as FI-modules
cites work: The structure of the Torelli group. III: The abelianization of \({\mathcal S}\)
cites work: Stability of Schur functors
cites work: Série de Poincaré-Koszul associée aux groupes de tresses pures
cites work: On the descending central series of groups with a single defining relation
cites work: On the homology and cohomology of congruence subgroups
cites work: On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes
cites work: The stable moduli space of Riemann surfaces: Mumford's conjecture
cites work: Q3997989
cites work: The Torelli groups for genus 2 and 3 surfaces
cites work: Q4486726
cites work: On the cohomology of compact homogeneous spaces of nilpotent Lie groups
cites work: Combinatorics and topology of complements of hyperplanes
cites work: Stability in the homology of congruence subgroups.
cites work: The Picard group of the moduli space of curves with level structures

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cites work: GL-equivariant modules over polynomial rings in infinitely many variables

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cites work: STABILITY PATTERNS IN REPRESENTATION THEORY

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cites work: Torsion in the Abelianization of the Torelli group \(\mathcal T\) as an \(\mathrm{Sp}(\mathbb Z_2)\)-module.

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cites work: Syzygies of Segre embeddings and \(\Delta\)-modules

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cites work: On the Steinberg Character of a Finite Simple Group of Lie Type

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cites work: Some aspects of groups acting on finite posets

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cites work: On the eigenvalues of representations of reflection groups and wreath products

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cites work: Q4550401

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cites work: Permutation representations on Schubert varieties

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cites work: Braid groups and loop spaces

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cites work: Q5490942

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cites work: Representation stability for the cohomology of the pure string motion groups.

Representation theory and homological stability. (DOI)
DOI: 10.1016/J.AIM.2013.06.016

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