representation, 2-representation, ∞-representation group, ∞-group group algebra, algebraic group, Lie algebra vector space, n-vector space affine space, symplectic vector space action, ∞-action module, equivariant object bimodule, Morita equivalence induced representation, Frobenius reciprocity Hilbert space, Banach space, Fourier transform, functional analysis orbit, coadjoint orbit, Killing form unitary representation geometric quantization, coherent state socle, quiver module algebra, comodule algebra, Hopf action, measuring 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 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...