Some Properties of Character Products

In practice the hypothesis that x(uv) # ~(u’v) for some conjugate U’ of u is almost always satisfied. When x is faithful and u = v-r, then this just asserts that u does not belong to the center of G (though it is an easy exercise to give a direct proof of the theorem in this case). We shall derive Theorem 1 from a general result (Theorem 2) about common constituents of permutation characters. This result is formulated in terms of characters of centralizer rings, and its proof is based upon the consideration of certain triple products of idempotents in centralizer rings. We have taken this opportunity to give a proof of the Krein condition announced in [9], in order to make the additional observation that the coefficients c,,~ which appear there also carry information on constituents of character products (Theorem 3). Again, this is in terms of characters of centralizer rings. As an application (Corollary 2) we prove a conjecture of Smith [lo, p. 231: If x, 5 are the nonprincipal of a primitive rank 3 group G of even order, then the restrictions of x, 5 to a point-stablilizer G, have a nonprincipal irreducible constituent in common. Also, we obtain under the same hypothesis a further condition on the Higman parameters (Corollary l), namely, the inequality E(l) -=z &x(l) (x(l) + 1). Since the appearance of [9], an excellent treatment of the Krein condition has been given by Higman in [5, 61. We have wholly adopted his point of view here, and our Theorem 3 should be regarded as a footnote to Higman’s theorem [5, Theorem (6.4)]. Finally it is worth noting for historical purposes that Higman’s approach is based on a theorem of Schur [S], which is even much older than Krein’s [7].