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

TL;DR

This paper explores the concept of representation stability within toric topology by analyzing $G$-polyhedral products derived from simplicial $G$-complexes, providing criteria for stability of symmetric group representations.

Contribution

It introduces a framework for studying representation stability of $G$-polyhedral products in toric topology, with specific criteria for symmetric groups.

Findings

Homotopy decomposition of $ ext{Sigma}(X, A)^K$ is $G$-equivariant after suspension.

Criteria established for simplicial $ ext{Sigma}_m$-complexes ensuring representation stability.

Provides conditions under which the homology representations stabilize across sequences.

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 $\Sigma (X, A)^K$ 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 $\{H_i((X, A)^{K_m})\}$.