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Representation Stability for Configuration Spaces of Graphs

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arxiv 1701.03490 v2 pith:SBYEPXXP submitted 2017-01-12 math.AT

Daniel Lütgehetmann This is my paper · ORCID
classification math.AT
keywords graphsconfigurationgrouprepresentationsequencespacessymmetricaction
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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 $q = 1$.

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  1. Hilbert Series for Configuration Spaces of Punctured Surfaces

    math.AG 2025-07 accept novelty 8.0 of 10

    For every genus g and one or more punctures, the mixed Hodge numbers of all unordered configuration spaces are encoded by a single rational generating function.

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