This paper studies the integral representations Vn and Vn′ of the symmetric groups &Sgr;n and &Sgr;n+1, respectively, given by the unique non-vanishing reduced integral homology of the space of fully-grown n-trees Tn, where &Sgr;n is the group of permutations of the set {1,...,n}.
Having provided a combinatorial description of these representations, the author shows various properties of the representations, and in particular that there is a short exact sequence of Z&Sgr;n+1-modules. Finally, the author proves that there is an isomorphism of Z&Sgr;n-modules
Lien ≊ Hom(Vn, Z[-1])
for the Lie representation of &Sgr;n, Lien, and the sign representation Z[-1].
