geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
The term representation stability [Church & Farb (2013)] refers to the phenomenon that for various sequences of group representations which arise in topology — notably for (symmetric group representations on) (co)homology groups of ordered configuration spaces of points and other moduli spaces — certain aspects, such as the multiplicities of irreducible sub-representations, eventually stabilize.
Specifically for the case of the ordinary homology of configuration spaces of points the phenomenon of representation stability is a refinement of the older notion of homological stabilization: Namely, where homological stabilization applies to un-ordered configurations of points, and says that the ordinary homology of these spaces eventually stabilizes (i.e. no longer increases) as the number of points grows, the same is far from true, at face value, for the ordered configuration spaces, the rank of whose homology groups instead increases strictly monotonically with the number of points. But the homology of unordered configuration spaces is still equipped with an action of the symmetric group $Sym(n)$ on the given number of points, and the statement of representation stability is that as symmetric group representations the homology still does stabilize, notably in that the number and multiplicity of irreducible representations stabilizes.
In particular, the homology of the un-ordered configuration spaces is canonically included into that of ordered configuration spaces as the summand of the trivial representation, and in this way representation stability does subsume and generalize the phenomenon of homological stabilization.
Moreover, it turns out that the general phenomenon and the details of representation stability for ordered configuration spaces of points is abstractly all governed by the fact that, as one looks at all numbers of points at once, their homology forms an FI-representation namely a functor from the category $FinSet_{inj}$ of finite sets with (just) injections between them.
The observation originates with:
245 (2013) 250-314 [doi:10.1016/j.aim.2013.06.016]
Lecture notes:
Introduction and review:
Benson Farb, Representation Stability [arXiv:1404.4065]
Jenny Wilson, A brief introduction to representation stability, Oberwolfach Workshop (Jan 2018) [pdf, pdf]
Jennifer C. H. Wilson, Representation stability and the braid groups, talk at ICERM – Braids (Feb 2022) [pdf]
Jenny Wilson, Stability patterns for braid groups and configuration spaces, talk at CQTS (Apr 2023) [web, slides:pdf, video:YT]
Rita Jimenez Rolland, Jennifer C. H. Wilson, Stability properties of moduli spaces, Notices of the AMS 69 4 (April 2022) [arXiv:2201.04096, pdf, web]
Expressing the rational cohomology of ordered configuration spaces of points via factorization homology and Ran spaces, and relation to higher representation stability:
In view of FI-representation-theory:
Trevor Hyde, $FI$-Modules and Representation Stability (2016) [pdf]
Nir Gadish, Categories of FI type: a unified approach to generalizing representation stability and character polynomials, Journal of Algebra 480 (2017) 450-486 [arXiv:1608.02664, doi:10.1016/j.jalgebra.2017.03.010]
Joe Moeller, Extensions of representation stable categories [arXiv:2209.03879]
See also:
Discussion of FI-representations in the generality of $\infty$-representations in stable $\infty$-categories and their analysis via Goodwillie calculus:
Last revised on September 5, 2023 at 13:56:40. See the history of this page for a list of all contributions to it.