Regular polytopes of rank $n/2$ for transitive groups of degree $n$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914076870836224 |
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| author | Fernandes, Maria Elisa Piedade, Claudio Alexandre |
| author_facet | Fernandes, Maria Elisa Piedade, Claudio Alexandre |
| contents | Previous research established that the maximal rank of the abstract regular polytopes whose automorphism group is a transitive proper subgroup of $\mbox{S}_n$ is $n/2 + 1$. Up to isomorphism and duality, when $n\geq 12$, there are only two polytopes attaining this rank and they occur when $n/2$ is odd, and hence have even rank.
In this paper, we investigate the case where the rank is equal to $n/2$ ($n\geq 14$). Our analysis suggests that reducing the rank by one results in a substantial increase in the number of regular polytopes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_16003 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regular polytopes of rank $n/2$ for transitive groups of degree $n$ Fernandes, Maria Elisa Piedade, Claudio Alexandre Combinatorics Group Theory 52B11, 20B35, 20B30, 05C25 Previous research established that the maximal rank of the abstract regular polytopes whose automorphism group is a transitive proper subgroup of $\mbox{S}_n$ is $n/2 + 1$. Up to isomorphism and duality, when $n\geq 12$, there are only two polytopes attaining this rank and they occur when $n/2$ is odd, and hence have even rank. In this paper, we investigate the case where the rank is equal to $n/2$ ($n\geq 14$). Our analysis suggests that reducing the rank by one results in a substantial increase in the number of regular polytopes. |
| title | Regular polytopes of rank $n/2$ for transitive groups of degree $n$ |
| topic | Combinatorics Group Theory 52B11, 20B35, 20B30, 05C25 |
| url | https://arxiv.org/abs/2407.16003 |