Character products and Q-polynomial group association schemes

We study a finite group having a faithful character whose square has a smal l number of irreducible characters as constituents. Let Irr(G) be the set o f absolutely irreducible ordinary characters of a finite group G. For each phi epsilon Irr(G), let <(phi)over cap> = phi if phi is real valued and <(p hi)over cap> = phi + <(phi)over bar> otherwise, where <(phi)over bar> denot es the complex conjugate of phi. Let RIrr(G) = {<(phi)over cap> \ phi Irr(G )}. For <(chi)over cap> epsilon RIrr(G) let <(chi)over cap>(2) = b. 1 + a . <(chi)over cap> + Psi such that Psi is a character of G which does not con tain chi nor the principal character 1 as a constituent. We study the case when Psi is a scalar multiple of a sum of the characters in RIrr(G), which are in a single orbit with respect to the action of the Galois group Gal((Q ) over bar/Q(<(chi)over cap>)). Here (Q) over bar denotes the algebraic clo sure of Q in C and Q(<(chi)over cap>) is the field generated by the values of <(chi)over cap>. As an application, we give a classification of Q-polyno mial group association schemes. (C) 2000 Academic Press.