Representation Stability for Families of Linear Subspace Arrangements

TL;DR

This paper extends the concept of representation stability from hyperplane arrangements to general linear subspace arrangements, introducing a framework for finitely generated arrangements and broadening stability results to various group actions.

Contribution

It introduces the notion of finitely generated arrangements and generalizes stability results to broader classes of arrangements and group actions, expanding the applicability of representation stability theory.

Findings

Representation stability applies to general linear subspace arrangements.

The theory of generalized character polynomials is developed for wide classes of groups.

Classical cohomological stability is achieved for quotients of linear arrangements.

Abstract

Church-Ellenberg-Farb used the language of FI-modules to prove that the cohomology of certain sequences of hyperplane arrangements with S_n-actions satisfies representation stability. Here we lift their results to the level of the arrangements themselves, and define when a collection of arrangements is "finitely generated". Using this notion we greatly widen the stability results to: 1) General linear subspace arrangements, not necessarily of hyperplanes. 2) A wide class of group actions, replacing FI by a general category C. We show that the cohomology of such collections of arrangements satisfies a strong form of representation stability, with many concrete applications. For this purpose we develop a theory of representation stability and generalized character polynomials for wide classes of groups. We apply this theory to get classical cohomological stability of quotients of linear subspace arrangements with coefficients in certain constructible sheaves.