A character theoretic formula for base size

Coen del Valle

doi:10.1007/s00013-025-02120-2

A character theoretic formula for base size

Coen del Valle^*

^*Corresponding author for this work

Pure Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

A base for a permutation group G acting on a set Ω is a sequence B of points of Ω such that the pointwise stabiliser G_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 ∈ ℕ. As a consequence of our results, we present a very short, entirely algebraic proof of the formula of Mecenero and Spiga for the base size of the symmetric group S_n acting on the k-element subsets of {1, 2, 3,...,n}. Our methods also provide a formula for the base size of many product type permutation groups.

Original language	English
Pages (from-to)	485-490
Number of pages	6
Journal	Archiv der Mathematik
Volume	124
Issue number	5
Early online date	24 Mar 2025
DOIs	https://doi.org/10.1007/s00013-025-02120-2
Publication status	E-pub ahead of print - 24 Mar 2025

Keywords

Base size
Irreducible character
Large base

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Cite this

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title = "A character theoretic formula for base size",

abstract = "A base for a permutation group G acting on a set Ω is a sequence B of points of Ω such that the pointwise stabiliser GB 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 ∈ ℕ. As a consequence of our results, we present a very short, entirely algebraic proof of the formula of Mecenero and Spiga for the base size of the symmetric group Sn acting on the k-element subsets of {1, 2, 3,...,n}. Our methods also provide a formula for the base size of many product type permutation groups.",

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author = "{del Valle}, Coen",

note = "Funding: Research of Coen del Valle is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), [funding reference number PGSD-577816-2023], as well as a University of St Andrews School of Mathematics and Statistics Scholarship.",

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AU - del Valle, Coen

N1 - Funding: Research of Coen del Valle is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), [funding reference number PGSD-577816-2023], as well as a University of St Andrews School of Mathematics and Statistics Scholarship.

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N2 - A base for a permutation group G acting on a set Ω is a sequence B of points of Ω such that the pointwise stabiliser GB 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 ∈ ℕ. As a consequence of our results, we present a very short, entirely algebraic proof of the formula of Mecenero and Spiga for the base size of the symmetric group Sn acting on the k-element subsets of {1, 2, 3,...,n}. Our methods also provide a formula for the base size of many product type permutation groups.

AB - A base for a permutation group G acting on a set Ω is a sequence B of points of Ω such that the pointwise stabiliser GB 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 ∈ ℕ. As a consequence of our results, we present a very short, entirely algebraic proof of the formula of Mecenero and Spiga for the base size of the symmetric group Sn acting on the k-element subsets of {1, 2, 3,...,n}. Our methods also provide a formula for the base size of many product type permutation groups.

KW - Base size

KW - Irreducible character

KW - Large base

U2 - 10.1007/s00013-025-02120-2

DO - 10.1007/s00013-025-02120-2

M3 - Article

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VL - 124

SP - 485

EP - 490

JO - Archiv der Mathematik

JF - Archiv der Mathematik

IS - 5

ER -

A character theoretic formula for base size

Abstract

Keywords

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