The ring of symmetric polynomials in n variables may be interpreted as a ring of characters of the general linear group GL(n) (see e.g. [4], §3·5). We consider here a generalization of symmetric function theory in which we deal with trace functions on several matrices over ℂ An important feature of the ‘single matrix variable’ (or classical) case is the characteristic isomorphism (see [5], chapter I, §7) ch: R = between the graded ring R, whose component Rd in degree d is the character group of the symmetric group of degree d, and the ring of symmetric functions Λ in infinitely many variables (obtained from symmetric functions in n variables by taking the graded inverse limit with respect to n).