Representation theory and homological stability

By Thomas Church, Benson Farb. 2010

277 citations recorded by the source corpus.

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 by Church–Ellenberg–Farb (in preparation) [20] to counting problems in number theory and finite group theory. Representation stability is also used by Church (2012) [19] to give broad generalizations and new proofs of classical homological stability theorems for configuration spaces on oriented manifolds.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bb288c7edf2018b06141e4163c98622c0322260e

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Citations in the corpus

Symmetric group characters as symmetric functions (2015, background): “refore f[Ξµ] = f[Ξµ] and g[Ξµ] = g[Ξµ] and hence f[Ξµ] = g[Ξµ]. By Proposition 54 we can conclude that f = g. 10. AppendixIII:ExamplesofSn modulesandtheir characters A recent paper by Church and Farb [CF] introduces a notion of representation stability. They describe a number of families of represen”
Stability in the homology of congruence subgroups (2012, background): “This notion of stability was first introduced by Church and Farb; see their paper [9] for many examples of situations “in nature” in which it occurs.”
Representation Stability (2014, methodology): “1 of [11]), we can apply an analogue of Theorem 5.”
Higher-order representation stability and ordered configuration spaces of manifolds (2016, background): “ this perspective, Theorem1.2becomes a structural algebraic result on the homology of conguration spaces. We now review elements of the theory of representation stability and FI-modules. Church{Farb [CF13] introduced the concept of representation stability to generalize homological stability to sequ”
GL-EQUIVARIANT MODULES OVER POLYNOMIAL RINGS IN INFINITELY MANY VARIABLES. II (2012, background): “ed. In particular, this shows that the local cohomology of M(Cn) with respect to determinantal ideals is ﬁnitely generated for n ≫ 0, and exhibits representation stability in the sense of Church–Farb [CF]. We also study the depth of M(Cn) with respect to a r(Cn) and show that, for n ≫ 0, it has the ”
Homological stability for configuration spaces of manifolds (2011, methodology): “However, in Church–Farb [CF] we defined representation stability, which lets us nevertheless give a meaningful answer to the question: “What is the stable homology of the ordered configuration space Cn(M)?” Representation stability is a notion which applies to a sequence of rational vector spaces Vn”
FI-modules: a new approach to stability for S_n-representations (2012, methodology): “58, relating stability degree of FI-modules with uniform representation stability in the sense of [CF].”
HOMOLOGICAL STABILITY FOR UNORDERED CONFIGURATION SPACES (2011, background): “Church has a preprint [3] in which he establishes rational homological stability in the representational sense (for the definition of which we refer to his paper with Farb [4]) of the ordered configuration spaces of a manifold, and this in particular implies Theorems B and C for the field Q.”
Homological stability for automorphism groups (2014, background): “On the level of groupoids, these maps define in each case a symmetric or braided monoidal structure.”
FI-modules and the cohomology of modular representations of symmetric groups (2015, background): “to make sense out of this we need coherent sequences of Sn-representations. The coherent sequences of Sn-representations were ﬁrst observed as a ubiquitously occurring phenomena by Church and Farb in [CF]. Soon after, the theory of FI-modules was introduced and developed by Church, Ellenberg and Far”
Correction to the article A spectral sequence for stratified spaces and configuration spaces of points (2016, background): “inson’s theorem [Be˘ı87], but I do not know if this is known and I have not attempted to prove it. 4. Representation stability The notion of representation stability was introduced by Church and Farb [CF13] as an extension of homological stability to situations where the Betti numbers do not actuall”
STABILITY PATTERNS IN REPRESENTATION THEORY (2013, background): “to be the case: we will see that these categories satisfy elegant universal properties and are closely related to Deligne’s interpolation categories [De2]. •Church, Ellenberg and Farb [CEF] (see also [CF]) give many examples of compatible sequences of representations of S(d) (which they call FI-modu”
FI-modules and stability for representations of symmetric groups (2012, background): “the notion of a stability degree for an FI-module over an arbitrary ring k. In characteristic 0, the stability degree provides a counterpart to the \stable range" for representation stability in [CF], as we prove later in Proposition 3.3.3. Coinvariants of FI-modules. If V n is a representation”
FI_W-modules and stability criteria for representations of the classical Weyl groups (2013, methodology): “2 Representation stability of finitely generated FIW–modules Prior to their work with Ellenberg on FI–modules, Church and Farb defined and developed the theory of representation stability for families of groups Gn including Sn and Bn [CF13].”
Representation stability in cohomology and asymptotics for families of varieties over finite fields (2013, background): “re well-understood, both on the topological and combinatorial side. Our focus in this paper is on a new kind of combinatorial stability, whose topological reﬂection is the representation stability of [CF] and [CEF]. The rest of the introduction will be spent explaining this connection. The new featu”
Representation stability for the cohomology of the pure string motion groups (2011, methodology): “The concept of representation stability was introduced by Church and Farb in [6].”
FI-modules over Noetherian rings (2012, background): “This turns out to be essentially equivalent to the phenomenon that was called “representation stability” in the earlier work of the first and third authors [7].”