A character theoretic formula for base size
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912041378250752 |
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| author | del Valle, Coen |
| author_facet | del Valle, Coen |
| contents | A base for a permutation group $G$ acting on a set $Ω$ is a sequence $\mathcal{B}$ of points of $Ω$ such that the pointwise stabiliser $G_{\mathcal{B}}$ is trivial. The base size of $G$ is the size of a smallest base for $G$. We derive a character theoretic formula for the base size of a class of groups admitting a certain kind of irreducible character. Moreover, we prove a formula for enumerating the non-equivalent bases for $G$ of size $l\in\mathbb{N}$. As a consequence of our results, we present a very short, entirely algebraic proof of the formula of Mecenero and Spiga~\cite{MeSp} for the base size of the symmetric group $\mathrm{S}_n$ acting on the $k$-element subsets of $\{1,2,3,\dots,n\}$. Our methods also provide a formula for the base size of many product-type permutation groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_15153 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A character theoretic formula for base size del Valle, Coen Group Theory Combinatorics A base for a permutation group $G$ acting on a set $Ω$ is a sequence $\mathcal{B}$ of points of $Ω$ such that the pointwise stabiliser $G_{\mathcal{B}}$ is trivial. The base size of $G$ is the size of a smallest base for $G$. We derive a character theoretic formula for the base size of a class of groups admitting a certain kind of irreducible character. Moreover, we prove a formula for enumerating the non-equivalent bases for $G$ of size $l\in\mathbb{N}$. As a consequence of our results, we present a very short, entirely algebraic proof of the formula of Mecenero and Spiga~\cite{MeSp} for the base size of the symmetric group $\mathrm{S}_n$ acting on the $k$-element subsets of $\{1,2,3,\dots,n\}$. Our methods also provide a formula for the base size of many product-type permutation groups. |
| title | A character theoretic formula for base size |
| topic | Group Theory Combinatorics |
| url | https://arxiv.org/abs/2409.15153 |