A character theoretic formula for base size

TL;DR

This paper develops a character theoretic approach to determine the minimal base size of certain permutation groups, providing new formulas and algebraic proofs for classical and product-type groups.

Contribution

It introduces a novel character theoretic formula for base size, applicable to groups with specific irreducible characters, and offers algebraic proofs for known and new results.

Findings

Derived a character theoretic formula for base size

Provided a formula for counting non-equivalent bases

Presented an algebraic proof for the symmetric group's base size

Abstract

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.