Strong Gelfand pairs of the sporadic groups and their extensions

TL;DR

This paper classifies strong Gelfand pairs involving sporadic groups, their automorphisms, and extensions, providing a comprehensive understanding of multiplicity-free induction properties in these complex finite groups.

Contribution

It identifies all strong Gelfand pairs for sporadic groups, their automorphism groups, and certain extensions, expanding the knowledge of their representation theory.

Findings

Complete list of strong Gelfand pairs for sporadic groups.

Identification of strong Gelfand pairs in Mathieu groups, Tits group, and Leech lattice automorphisms.

New insights into the structure of multiplicity-free inductions in finite groups.

Abstract

A strong Gelfand pair $(G, H)$ is a finite group $G$ and a subgroup $H$ where every irreducible character of $H$ induces to a multiplicity-free character of $G$. We determine the strong Gelfand pairs of the sporadic groups, their automorphism groups, and their covering groups. We also find the (strong) Gelfand pairs of the generalized Mathieu groups, the Tits group, and the automorphism group of the Leech Lattice.