Symmetric functions in noncommuting variables

Abstract. Consider the algebra Q〈〈x1, x2,...〉 〉 of formal power series in countably many noncommuting variables over the rationals. The subalgebra Π(x1, x2,...) of symmetric functions in noncommuting variables consists of all elements invariant under permutation of the variables and of bounded degree. We develop a theory of such functions analogous to the ordinary theory of symmetric functions. In particular, we define analogs of the monomial, power sum, elementary, complete homogeneous, and Schur symmetric functions as well as investigating their properties. 1