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Representation theory and homological stability. (English) Zbl 1300.20051
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 Torelli surface groups is representation stable with period 2, and the Abelianization of the corresponding level \(p\) mapping class groups has the same property, with period \(p\).

MSC:
20G05 Representation theory for linear algebraic groups
20G10 Cohomology theory for linear algebraic groups
20J06 Cohomology of groups
20F36 Braid groups; Artin groups
19B14 Stability for linear groups
55R80 Discriminantal varieties and configuration spaces in algebraic topology
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