A character theoretic formula for base size
| Autor: | del Valle, Coen |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | A base for a permutation group $G$ acting on a set $\Omega$ is a sequence $\mathcal{B}$ of points of $\Omega$ 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. Comment: 4 pages |
| Databáze: | arXiv |
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