Cameron-Liebler sets in permutation groups

TL;DR

This paper explores Cameron-Liebler sets within permutation groups, especially 2-transitive groups, providing new constructions and insights into their algebraic structure.

Contribution

It introduces the concept of Cameron-Liebler sets in permutation groups and develops methods for constructing such sets in 2-transitive groups.

Findings

New constructions of Cameron-Liebler sets for 2-transitive groups

Characterization of Cameron-Liebler sets in permutation groups

Insights into the algebraic structure of these sets

Abstract

Consider a group $G$ acting on a set $\Omega$, the vector $v_{a,b}$ is a vector with the entries indexed by the elements of $G$, and the $g$-entry is 1 if $g$ maps $a$ to $b$, and zero otherwise. A $(G,\Omega)$-Cameron-Liebler set is a subset of $G$, whose indicator function is a linear combination of elements in $\{v_{a, b}\ :\ a, b \in \Omega\}$. We investigate Cameron-Liebler sets in permutation groups, with a focus on constructions of Cameron-Liebler sets for 2-transitive groups.