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 (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 Σm–polyhedral products, we give criteria on simplicial Σm–complexes which imply representation stability of Σm–representations {Hi((X,A)Km)}.

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