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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.14176 |
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| _version_ | 1866909915346370560 |
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| author | Lee, Seunghun Nevo, Eran |
| author_facet | Lee, Seunghun Nevo, Eran |
| contents | We show that for $2\le d\le 4$, every finite geometric simplicial complex $Δ$ in $\mathbb{R}^d$ with vertices on the moment curve can be extended to a triangulation $T$ of the cyclic polytope $C$ where $Δ, T$ and $C$ all have the same vertex set. Further, for $d\ge 5$ we construct for every $n\ge d+3$ complexes $Δ$ on $n$ vertices for which no such triangulations $T$ exist.
Our result for $d=4$ has the following novel algebraic application, due to a correspondence by Oppermann and Thomas (JEMS, 2012): every maximal rigid object in $\mathcal{O}_{A_n^{2}}$ is cluster tilting, where $\mathcal{O}_{A_n^δ}$ denotes a higher dimensional cluster category introduced by Oppermann and Thomas for $A_n^δ$, where $A_n^δ$ denotes a higher Auslander algebra of linearly oriented type $A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_14176 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On an extension problem on the moment curve Lee, Seunghun Nevo, Eran Combinatorics 52B05, 52B12, 05E45, 16G10, 13F60 We show that for $2\le d\le 4$, every finite geometric simplicial complex $Δ$ in $\mathbb{R}^d$ with vertices on the moment curve can be extended to a triangulation $T$ of the cyclic polytope $C$ where $Δ, T$ and $C$ all have the same vertex set. Further, for $d\ge 5$ we construct for every $n\ge d+3$ complexes $Δ$ on $n$ vertices for which no such triangulations $T$ exist. Our result for $d=4$ has the following novel algebraic application, due to a correspondence by Oppermann and Thomas (JEMS, 2012): every maximal rigid object in $\mathcal{O}_{A_n^{2}}$ is cluster tilting, where $\mathcal{O}_{A_n^δ}$ denotes a higher dimensional cluster category introduced by Oppermann and Thomas for $A_n^δ$, where $A_n^δ$ denotes a higher Auslander algebra of linearly oriented type $A$. |
| title | On an extension problem on the moment curve |
| topic | Combinatorics 52B05, 52B12, 05E45, 16G10, 13F60 |
| url | https://arxiv.org/abs/2511.14176 |