Representation Stability for Configuration Spaces of Graphs

Abstract

We consider for two based graphs $G$ and $H$ the sequence of graphs $G_k$ given by the wedge sum of $G$ and $k$ copies of $H$. These graphs have an action of the symmetric group ${\Sigma }_k$ by permuting the $H$-summands. We show that the sequence of representations of the symmetric group $H_q({\mathrm{Conf}}_n(G_{\bullet });\mathbf{Q})$, the homology of the ordered configuration space of these spaces, is representation stable in the sense of Church and Farb. In the case where $G$ and $H$ are trees, we provide a similar result for glueing along arbitrary subtrees instead of the base point. Furthermore, we show that stabilization alway holds for $q=1$.

arXiv commentMajor rewrite

Cited by in corpus (2)

• Non-abelian Quantum Statistics on Graphs
• Edge stabilization in the homology of graph braid groups