Simplicial G–complexes and representation stability of polyhedral products

Fu, Xin; Grbić, Jelena

doi:10.2140/agt.2020.20.215

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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

DOI: 10.2140/agt.2020.20.215

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 $(X, A)$ , a $G$ –polyhedral product ${(X, A)}^{K}$ is associated. We show that the homotopy decomposition 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 ${H_{i} ({(X, A)}^{K_{m}})}$ .

Keywords

polyhedral products, representation stability, symmetric groups

Mathematical Subject Classification 2010

Primary: 20C30

Secondary: 05E10, 55N91, 55U10

References

Publication

Received: 5 March 2018

Revised: 17 November 2018

Accepted: 24 February 2019

Published: 23 February 2020

Authors

Xin Fu
School of Mathematics University of Southampton Southampton United Kingdom

Jelena Grbić
School of Mathematics University of Southampton Southampton United Kingdom

Volume 20, issue 1 (2020)