An acyclic $d$-partition of the $r$-uniform complete hypergraph $K_{rd}^{(r)}$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915364186619904 |
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| author | Carter, Ayako Montoya, Eric Staic, Mihai D. |
| author_facet | Carter, Ayako Montoya, Eric Staic, Mihai D. |
| contents | In this paper we introduce a $d$-partition $\mathcal{E}_d^{(r)}=(Ω_1^{(r,d)}, Ω_2^{(r,d)},\dots, Ω_d^{(r,d)})$ of the $r$-uniform complete hypergraph $K_{rd}^{(r)}$. We prove that $\mathcal{E}_d^{(r)}$ is homogeneous and that each hypergraph $Ω_i^{(r,d)}$ is acyclic (i.e. has zero Betti numbers). As an application, we show that the map $det^{S^r}$ is nontrivial for every $r$, which gives a partial answer to a conjecture from [14]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_23238 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An acyclic $d$-partition of the $r$-uniform complete hypergraph $K_{rd}^{(r)}$ Carter, Ayako Montoya, Eric Staic, Mihai D. Combinatorics Primary 05C65 In this paper we introduce a $d$-partition $\mathcal{E}_d^{(r)}=(Ω_1^{(r,d)}, Ω_2^{(r,d)},\dots, Ω_d^{(r,d)})$ of the $r$-uniform complete hypergraph $K_{rd}^{(r)}$. We prove that $\mathcal{E}_d^{(r)}$ is homogeneous and that each hypergraph $Ω_i^{(r,d)}$ is acyclic (i.e. has zero Betti numbers). As an application, we show that the map $det^{S^r}$ is nontrivial for every $r$, which gives a partial answer to a conjecture from [14]. |
| title | An acyclic $d$-partition of the $r$-uniform complete hypergraph $K_{rd}^{(r)}$ |
| topic | Combinatorics Primary 05C65 |
| url | https://arxiv.org/abs/2506.23238 |