Branching multiplicities for arbitrary cyclic subgroups of finite groups

Determine a combinatorial description of the multiplicities ⟨Res^G_H χ, ψ⟩ and characterize their nonvanishing support for restrictions of irreducible representations χ of an arbitrary finite group G to irreducible characters ψ of an arbitrary cyclic subgroup H ≤ G (not necessarily of prime order), extending the known results for cyclic subgroups of prime order.

Background

The paper surveys prior work on restriction problems focused largely on cyclic subgroups of prime order across various families of groups, noting extensive results in those cases. By contrast, it emphasizes that analogous problems for cyclic subgroups of composite order are not well understood in general finite groups.

The authors then focus their new contributions on dihedral subgroups of the symmetric and alternating groups, while explicitly remarking that the broader problem for arbitrary cyclic subgroups of finite groups remains unresolved, thereby situating their results within this open context.