ON SYMMETRICAL 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 greater than or equal to 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). (C) 1995 Academic Press, Inc.