Association Schemes with at Most Two Nonlinear Irreducible Characters and Applications to Finite Groups

An irreducible character of an association scheme is called nonlinear if the multiplicity of is greater than 1. The main result of this paper gives a characterization of commutative association schemes with at most two nonlinear irreducible characters. This yields a characterization of finite groups with at most two nonlinear irreducible characters. A class of noncommutative association schemes with at most two nonlinear irreducible character is also given.