Representation stability in the (co)homology of vertical configuration spaces
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910724170711040 |
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| author | Baron, David Pal, Urshita Wang, Chenglu Wilson, Jennifer C. H. Yang, Chunye |
| author_facet | Baron, David Pal, Urshita Wang, Chenglu Wilson, Jennifer C. H. Yang, Chunye |
| contents | In this paper, we study sequences of topological spaces called "vertical configuration spaces" of points in Euclidean space. We apply the theory of FI$_G$-modules, and results of Bianchi-Kranhold, to show that their (co)homology groups are "representation stable" with respect to natural actions of wreath products $S_k \wr S_n$. In particular, we show that in each (co)homological degree, the (co)homology groups (viewed as $S_k \wr S_n$-representations) can be expressed as induced representations of a specific form. Consequently, the characters of their rational (co)homology groups, and the patterns of irreducible $S_k \wr S_n$-representation constituents of these groups, stabilize in a strong sense. In addition, we give a new proof of rational (co)homological stability for unordered vertical configuration spaces, with an improved stable range. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_01128 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Representation stability in the (co)homology of vertical configuration spaces Baron, David Pal, Urshita Wang, Chenglu Wilson, Jennifer C. H. Yang, Chunye Algebraic Topology Combinatorics 55R80, 18A25, 55R40, 20E22, 05E10 In this paper, we study sequences of topological spaces called "vertical configuration spaces" of points in Euclidean space. We apply the theory of FI$_G$-modules, and results of Bianchi-Kranhold, to show that their (co)homology groups are "representation stable" with respect to natural actions of wreath products $S_k \wr S_n$. In particular, we show that in each (co)homological degree, the (co)homology groups (viewed as $S_k \wr S_n$-representations) can be expressed as induced representations of a specific form. Consequently, the characters of their rational (co)homology groups, and the patterns of irreducible $S_k \wr S_n$-representation constituents of these groups, stabilize in a strong sense. In addition, we give a new proof of rational (co)homological stability for unordered vertical configuration spaces, with an improved stable range. |
| title | Representation stability in the (co)homology of vertical configuration spaces |
| topic | Algebraic Topology Combinatorics 55R80, 18A25, 55R40, 20E22, 05E10 |
| url | https://arxiv.org/abs/2412.01128 |