Strong Gelfand pairs of the sporadic groups and their extensions

TL;DR

This work delivers a complete map of strong Gelfand pairs for the sporadic groups, their automorphism and covering groups, and related families such as the generalized Mathieu groups, the Leech lattice automorphism group, and the Tits group. It combines theoretical reduction lemmas with exhaustive computations in GAP and MAGMA to certify multiplicity-free inductions and to extend results through primes and abelian extensions. The paper provides explicit lists of all strong Gelfand pairs in these families, together with structural insights into extension phenomena and a substantial set of accompanying code for verification. These results deepen understanding of multiplicity-free representation theory in large finite groups and furnish practical data resources for algebraic combinatorics and related areas.

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.

Theorems & Definitions (22)

• Lemma 1.1
• Lemma 1.2
• Theorem 2.1
• proof
• Theorem 3.1
• Theorem 3.2
• proof
• Corollary 3.1
• Theorem 3.3
• proof