Two gluing methods for string C-group representations of the symmetric groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908336476127232 |
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| author | Leemans, Dimitri Mulpas, Jessica |
| author_facet | Leemans, Dimitri Mulpas, Jessica |
| contents | The study of string C-group representations of rank at least $n/2$ for the symmetric group $S_n$ has gained a lot of attention in the last fifteen years. In a recent paper, Cameron et al. gave a list of permutation representation graphs of rank $r\geq n/2$ for $S_n$, having a fracture graph and a non-perfect split. They conjecture that these graphs are permutation representation graphs of string C-groups. In trying to prove this conjecture, we discovered two new techniques to glue two CPR graphs for symmetric groups together. We discuss the cases in which they yield new CPR graphs. By doing so, we invalidate the conjecture of Cameron et al. We believe our gluing techniques will be useful in the study of string C-group representations of high ranks for the symmetric groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_17535 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Two gluing methods for string C-group representations of the symmetric groups Leemans, Dimitri Mulpas, Jessica Combinatorics Group Theory 20B30, 05C25, 52B15 The study of string C-group representations of rank at least $n/2$ for the symmetric group $S_n$ has gained a lot of attention in the last fifteen years. In a recent paper, Cameron et al. gave a list of permutation representation graphs of rank $r\geq n/2$ for $S_n$, having a fracture graph and a non-perfect split. They conjecture that these graphs are permutation representation graphs of string C-groups. In trying to prove this conjecture, we discovered two new techniques to glue two CPR graphs for symmetric groups together. We discuss the cases in which they yield new CPR graphs. By doing so, we invalidate the conjecture of Cameron et al. We believe our gluing techniques will be useful in the study of string C-group representations of high ranks for the symmetric groups. |
| title | Two gluing methods for string C-group representations of the symmetric groups |
| topic | Combinatorics Group Theory 20B30, 05C25, 52B15 |
| url | https://arxiv.org/abs/2504.17535 |