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 dimHomG(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.