A character theoretic formula for base size

2024/09/23 by Coen del Valle, del Valle, Coen
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2409.15153

openalex publication_date 2024/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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~\citeMeSp 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.

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