Representation stability in the (co)homology of vertical configuration spaces

TL;DR

This paper demonstrates that the (co)homology groups of vertical configuration spaces exhibit representation stability under wreath product actions, with explicit descriptions and improved stability ranges, using FI$_G$-modules theory.

Contribution

It introduces a new proof of rational (co)homological stability for vertical configuration spaces with enhanced stable ranges and describes their (co)homology as induced representations, advancing understanding of their structure.

Findings

(Co)homology groups are representation stable as $S_k times S_n$-representations.

Characters and irreducible constituents stabilize strongly.

Provides an improved proof with better stable range for stability.

Abstract

In this paper, we study sequences of topological spaces called "vertical configuration spaces" of points in Euclidean space. We apply the theory of FI$_G$-modules, and results of Bianchi-Kranhold, to show that their (co)homology groups are "representation stable" with respect to natural actions of wreath products $S_k \wr S_n$. In particular, we show that in each (co)homological degree, the (co)homology groups (viewed as $S_k \wr S_n$-representations) can be expressed as induced representations of a specific form. Consequently, the characters of their rational (co)homology groups, and the patterns of irreducible $S_k \wr S_n$-representation constituents of these groups, stabilize in a strong sense. In addition, we give a new proof of rational (co)homological stability for unordered vertical configuration spaces, with an improved stable range.