Simplicial $G$-complexes and representation stability of polyhedral products

Abstract

Representation stability in the sense of Church-Farb is concerned with stable properties of representations of sequences of algebraic structures, in particular of groups. We study this notion on objects arising in toric topology. With a simplicial $G$-complex $K$ and a topological pair $(X,A)$, a $G$-polyhedral product $(X,A{)}^K$ is associated. We show that the homotopy decomposition [2] of $\hat{I}\pounds (X,A{)}^K$ is then $G$-equivariant after suspension. In the case of $\hat{I}{\pounds }_m$-polyhedral products, we give criteria on simplicial $\hat{I}{\pounds }_m$-complexes which imply representation stability of $\hat{I}{\pounds }_m$-representations $\{H_i((X,A{)}^{K_m})\}$.

To appear in Algebraic & Geometric Topology