#### Volume 20, issue 1 (2020)

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

### Xin Fu and Jelena Grbić

Algebraic & Geometric Topology 20 (2020) 215–238
##### Abstract

Representation stability in the sense of Church and 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 $\left(X,A\right)$, a $G$–polyhedral product ${\left(X,A\right)}^{K}$ is associated. We show that the homotopy decomposition of $\Sigma {\left(X,A\right)}^{K}$ is then $G$–equivariant after suspension. In the case of ${\Sigma }_{m}$–polyhedral products, we give criteria on simplicial ${\Sigma }_{m}$–complexes which imply representation stability of ${\Sigma }_{m}$–representations $\left\{{H}_{i}\left({\left(X,A\right)}^{{K}_{m}}\right)\right\}$.

##### Keywords
polyhedral products, representation stability, symmetric groups
##### Mathematical Subject Classification 2010
Primary: 20C30
Secondary: 05E10, 55N91, 55U10