Multiplicity upon restriction to the derived subgroup

TL;DR

This paper proposes a conjecture about the multiplicity of irreducible subgroup representations within group representations, explores its implications, verifies some consequences, and provides examples and theorems related to multiplicity behavior.

Contribution

It introduces a new conjecture on representation multiplicities for groups with identical derived subgroups and verifies some of its consequences and specific cases.

Findings

Explicit example of multiplicity 2 upon restriction

Theorems showing multiplicity 1 in classical groups

Verification of some conjecture consequences

Abstract

We present a conjecture on multiplicity of irreducible representations of a subgroup $H$ contained in the irreducible representations of a group $G$, with $G$ and $H$ having the same derived groups. We point out some consequences of the conjecture, and verification of some of the consequences. We give an explicit example of multiplicity $2$ upon restriction, as well as certain theorems in the context of classical groups where the multiplicity is $1$.