Representation stability on the cohomology of complements of subspace arrangements

TL;DR

This paper investigates how the cohomology groups of complements of linear subspace arrangements stabilize as symmetric group representations, providing bounds and alternative proofs for when this stabilization occurs.

Contribution

It offers new bounds on the stabilization point and an alternative proof for representation stability in the cohomology of subspace arrangement complements.

Findings

Bounds on the point of stabilization for cohomology groups.

An alternative proof of the stabilization phenomenon.

Extension of known results to more general 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.