A character theoretic formula for base size II
Coen del Valle
TL;DR
This work addresses the base size problem for finite permutation groups by linking base-controlling irreducible characters with the Külshammer graph. It establishes a general character-theoretic formula $b(G)=\mathrm{Diam}(\mathcal{K}(G,H))+1=d(1_H,\phi\downarrow_H)+1$ for groups admitting a base-controlling homomorphism $\phi$, thereby settling the Fritzsche--Külshammer--Reiche conjecture for $G=\mathrm{S}_{n,k}$ and providing a broader framework beyond $S_{n,k}$. The approach unifies permutation-character methods with graph-theoretic paths in $\mathcal{K}(G,H)$, and yields a third, distinct formula for the base size of $\mathrm{S}_{n,k}$. Concrete examples, such as $\mathrm{PGL}_2(7)$ and dihedral groups, illustrate the applicability and show how the diameter of $\mathcal{K}(G,H)$ determines the base size via the base-controlling homomorphism $\phi$.
Abstract
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 $G$. Extending the results of a recent paper of the author, we prove a 2013 conjecture of Fritzsche, Külshammer, and Reiche. Moreover, we generalise this conjecture and derive an alternative character theoretic formula for the base size of a certain class of permutation groups. As a consequence of our work, a third formula for the base size of the symmetric group of degree $n$ acting on the subsets of $\{1,2,\dots, n\}$ is obtained.