Character Products and Q-Polynomial Group Association Schemes

We study a finite group having a faithful character whose square has a small number of irreducible characters as constituents. Let Irr(G) be the set of absolutely irreducible ordinary characters of a finite group G. For each φ Irr(G), let = φ if φ is real valued and = φ + otherwise, where denotes the complex conjugate of φ. Let RIrr(G) = { φ Irr(G)}. For RIrr(G), let such that Ψ is a character of G which does not contain χ nor the principal character 1 as a constituent. We study the case when Ψ is a scalar multiple of a sum of the characters in IRrr(G), which are in a single orbit with respect to the action of the Galois group Gal( /Q( )). Here denotes the algebraic closure of Q in C and Q( ) is the field generated by the values of . As an application, we give a classification of Q-polynomial group association schemes.