Free lie superalgebras and the representations of gl(m,n) and q(n)

Let L be the free Lie superalgebra generated by a Z(2)-graded vector space V over C. Suppose that g is a Lie superalgebra gl(m, n) or q (n). We study the g-module structure on the kth homogeneous component L-k of L when V is the natural representation of g. We give the multiplicities of irreducible representations of g in L-k by using the character of L-k. The multiplicities are given in terms of the character values of irreducible (projective) representations of the symmetric groups.