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

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 $Σ(X, A)^K$ is then $G$-equivariant after suspension. In the case of $Σ_m$-polyhedral products, we give criteria on simplicial $Σ_m$-complexes which imply representation stability of $Σ_m$-representations $\{H_i((X, A)^{K_m})\}$.

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