We give a new categorical way to construct the central stability homology of Putman and Sam and explain how it can be used in the context of representation stability and homological stability. In contrast to them, we cover categories with infinite automorphism groups. We also connect central stability homology to Randal-Williams and Wahl's work on homological stability. We also develop a criterion that implies that functors that are polynomial in the sense of Randal-Williams and Wahl are centrally stable in the sense of Putman.

This paper investigates the homology of the Brauer algebras, interpreted as appropriate $${{\,\mathrm{Tor}\,}}$$ Tor -groups, and shows that it is closely related to the homology of the symmetric group. Our main results show that when the defining parameter $$\delta $$ δ of the Brauer algebra is invertible, then the homology of the Brauer algebra is isomorphic to the homology of the symmetric group, and that when $$\delta $$ δ is not invertible, this isomorphism still holds in a range of degrees that increases with n.

We show, finitely generated rational VIC Q -modules and SI Q -modules are uniformly representation stable and all their submodules are finitely generated. We use this to prove two conjectures of Church and Farb, which state that the quotients of the lower central series of the Torelli subgroups of Aut (Fn) and Mod(Σ g,1 ) are uniformly representation stable as sequences of representations of the general linear groups and the symplectic groups, respectively. Furthermore we prove an analogous statement for their Johnson filtrations. PETER PATZTInjectivity: The map φ n : V n → V n+1 is injective for all large enough n ∈ N.Multiplicity stability: We can write. . ) and cλ ,n is independent of n for all large enough n ∈ N.A consistent sequence is called uniformly representation stable if the multiplicities cλ ,n stabilize uniformly.Functors from a category C to the category Q−mod of vector spaces over Q are called C-modules. Every FI-module V : FI → Q−mod gives rise to a consistent sequence, by takingThe connection to representation stability was provided by Church-Ellenberg-Farb in the following theorem.Theorem (Church-Ellenberg-Farb [CEF15, Thm 1.13]). An FI-module V is finitely generated if and only if its consistent sequence is uniformly representation stable and V n is finite dimensional for all n ∈ N.This theorem depends on the following noetherian property of FI-modules.Theorem (Church-Ellenberg-Farb [CEF15, Thm 1.3]). Every submodule of a finitely generated FI-module is finitely generated.Analogous theorems for the hyperoctahedral groups were proved by Wilson [Wil14, Thm 4.21 + Thm 4.22].Representation stability over the general linear groups and symplectic groups. The rational representation theory for both GL n Q and Sp 2n Q is semisimple and the irreducibles are indexed by pairs of partitions (λ + , λ − ) such that the lengths ℓ(λ + ) + ℓ(λ − ) ≤ n and by partitions λ whose length ℓ(λ) ≤ n, respectively. We respectively denote these irreducibles by GL n (λ + , λ − ) and Sp 2n (λ).For a consistent sequence of rational representations of the general linear groups or the symplectic groups Church-Farb [CF13, Def 2.3] define (uniform) representation stability analogously to the symmetric groups-only the analogue of the multiplicity stability condition is easier to state: ematical School, the SFB Raum-Zeit-Materie and the Dahlem Research School. The author also wants to thank for helpful conversations. Special thanks to Steven Sam for his extensive help with the modification rules, and to Kevin Casto for pointing out the conjectures to the author. Rational representation theory of the general linear groups and the symplectic groupsLet us start by shortly recalling the rational (or algebraic) representation theory of the algebraic groups GL n Q and Sp 2n Q. More elaboration can be found in the

No abstract