Representation Stability for Families of Linear Subspace Arrangements

2016/03/28 by Nir Gadish, Gadish, Nir
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Representation Theory (math.RT) #math.AG #math.AT #math.GT #math.RT

paper · pdf · doi:10.48550/arxiv.1603.08547

36 pages; this is a split version of the previously posted preprint - containing the topological aspects of the theory; the forthcoming second part will contain the rep theory aspects

arxiv created 2016/06/10 · arxiv updated 2016/06/13

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.

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