McKay trees

Given a finite group G and its representation ρ, the corresponding McKay graph is a graph Γ(G, ρ) whose vertices are the irreducible representations of G; the number of edges between two vertices π, τ of Γ(G, ρ) is dim Hom_G(π ⊗ ρ, τ). The collection of all McKay graphs for a given group G encodes, in a sense, its character table. Such graphs were also used by McKay to provide a bijection between the finite subgroups of SU(2) and the affine Dynkin diagrams of types A, D, E, the bijection given by considering the appropriate McKay graphs. In this paper, we classify all (undirected) trees which are McKay graphs of finite groups and describe the corresponding pairs (G, ρ); this classification turns out to be very concise. Moreover, we give a partial classification of McKay graphs which are forests, and construct some non-trivial examples of such forests.