Representation stability on the cohomology of complements of subspace arrangements

Abstract

We study representation stability in the sense of Church and Farb of sequences of cohomology groups of complements of arrangements of linear subspaces in real and complex space as $S_n$-modules. We consider arrangement of linear subspaces defined by sets of diagonal equalities $x_i = x_j$ and invariant under the action of $S_n$ permuting the coordinates. We provide bounds on the point when stabilization occurs and an alternative proof for the fact that stabilization happens. The latter is a special case of a very general stabilization result of Gadish and for the pure braid space the result is part of the work of Church and Farb. For this space better stabilization bounds were obtained by Hersh and Reiner.