Representation stability for the cohomology of arrangements associated to root systems

TL;DR

This paper proves that the rational cohomology of arrangements derived from root systems of types A, B, C, and D stabilizes as Weyl group representations, using spectral sequences and FI_W-module theory.

Contribution

It extends the concept of representation stability to cohomology of arrangements associated with root systems, combining spectral sequences and combinatorial poset descriptions.

Findings

Cohomology stabilizes as Weyl group representations for these arrangements.

Uses a combination of spectral sequence techniques and FI_W-module theory.

Provides a combinatorial description of intersection posets with labelled partitions.

Abstract

From a root system, one may consider the arrangement of reflecting hyperplanes, as well as its toric and elliptic analogues. The corresponding Weyl group acts on the complement of the arrangement and hence on its cohomology. We consider a sequence of linear, toric, or elliptic arrangements which arise from a family of root systems of type A, B, C, or D, and we show that the rational cohomology stabilizes as a sequence of Weyl group representations. Our techniques combine a Leray spectral sequence argument similar to that of Church in the type A case along with FI$_W$-module theory which Wilson developed and used in the linear case. A key to the proof relies on a combinatorial description, using labelled partitions, of the poset of connected components of intersections of subvarieties in the arrangement.