Transitivity in wreath products with symmetric groups
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908981305278464 |
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| author | Klawuhn, Lukas Schmidt, Kai-Uwe |
| author_facet | Klawuhn, Lukas Schmidt, Kai-Uwe |
| contents | It is known that the notion of a transitive subgroup of a permutation group $P$ extends naturally to the subsets of $P$. We study transitive subsets of the wreath product $G \wr S_n$, where $G$ is a finite abelian group. This includes the hyperoctahedral group for $G=C_2$. We give structural characterisations of transitive subsets using the character theory of $G \wr S_n$ and interpret such subsets as designs in the conjugacy class association scheme of $G \wr S_n$. In particular, we prove a generalisation of the Livingstone-Wagner theorem and give explicit constructions of transitive sets. Moreover, we establish connections to orthogonal polynomials, namely the Charlier polynomials, and use them to study codes and designs in $C_r \wr S_n$. Many of our results extend results about the symmetric group $S_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_20495 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Transitivity in wreath products with symmetric groups Klawuhn, Lukas Schmidt, Kai-Uwe Combinatorics Group Theory Representation Theory 05B99, 05E30, 20C99 It is known that the notion of a transitive subgroup of a permutation group $P$ extends naturally to the subsets of $P$. We study transitive subsets of the wreath product $G \wr S_n$, where $G$ is a finite abelian group. This includes the hyperoctahedral group for $G=C_2$. We give structural characterisations of transitive subsets using the character theory of $G \wr S_n$ and interpret such subsets as designs in the conjugacy class association scheme of $G \wr S_n$. In particular, we prove a generalisation of the Livingstone-Wagner theorem and give explicit constructions of transitive sets. Moreover, we establish connections to orthogonal polynomials, namely the Charlier polynomials, and use them to study codes and designs in $C_r \wr S_n$. Many of our results extend results about the symmetric group $S_n$. |
| title | Transitivity in wreath products with symmetric groups |
| topic | Combinatorics Group Theory Representation Theory 05B99, 05E30, 20C99 |
| url | https://arxiv.org/abs/2409.20495 |