Association schemes with at most two nonlinear irreducible characters and applications to finite groups

TL;DR

This paper characterizes commutative association schemes and finite groups with at most two nonlinear irreducible characters, and introduces a class of noncommutative schemes with similar properties, advancing understanding of their algebraic structure.

Contribution

It provides a new characterization of association schemes and finite groups based on the number of nonlinear irreducible characters, including noncommutative cases.

Findings

Characterization of commutative association schemes with ≤2 nonlinear irreducible characters

Characterization of finite groups with ≤2 nonlinear irreducible characters

Introduction of noncommutative association schemes with ≤2 nonlinear irreducible characters

Abstract

An irreducible character $\chi$ of an association scheme is called nonlinear if the multiplicity of $\chi$ 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.