On Symmetric Functions Related to Witt Vectors and the Free Lie Algebra

With the notations of Macdonald we define symmetric functions q_n by Eq. (2.1). We conjecture that for n≥2, −q_n is a sum of Schur functions and thus is the characteristic function of some representation of S_n. A first result is the "orthogonality relation" of Theorem 3.1, where l_n is the symmetric function corresponding to the nth free Lie algebra representation. The conjecture is deduced when n is a power of 2 (Corollary 3.5). When n is odd, a Hall basis construction shows that −q_n has positive coefficients (Corollary 4.7); when n is a power of an odd prime, the construction of a functor embedded in the free Lie algebra implies the conjecture in this case (Corollary 5.2).