The Tree Representation Of

We show that the space of fully-grown n-trees has the homotopy type of a bouquet of spheres of dimension n ? 3 and that the character of the representation of n+1 on its only non-trivial reduced homology group is (Ind n+1 n Lie n ?Lie n+1): Introduction We investigate a certain space T n called the space of fully-grown n-trees, and we prove that it has the homotopy type of a bouquet of spheres of dimension n ? 3. The symmetric group n+1 acts in a natural way on T n , so the reduced homology of T n aaords an integral representation of n+1 , which we call the tree representation. It has dimension (n ? 1)!. Its restriction to n?1 is isomorphic (over the integers) to the regular representation. We show that the character of the tree representation is (Ind n+1 n Lie n ?Lie n+1) where is the alternating character, and Lie n is the character of the representation of n on that component of the free Lie algebra on n generators which is spanned by those words involving each generator exactly once. In the proof we use the Lefschetz xed-point theorem, and a formula of Brandt 3] to the eeect that Lie r = Ind r C r r , where r is a faithful linear character of the subgroup generated by an r-cycle in r. We became interested in the space T n because it arises in ?-homology theory for commutative rings. This is very close to the study of E 1 structures in ho-motopy theory, where T n originated 2], and to connguration spaces and free Lie algebras. The connections among these are described in 5]. Restricting the tree representation to n gives Lie n. Our Theorem 3.1 therefore implies a result on the top-dimensional homology of connguration spaces due to F. Cohen (see 6], Thm.12.3, p.302) which is also proved by completely diierent methods in 9]. The restriction to n also yields a known result, due to Stanley 11] on the homology of partition lattices: there is an explicit proof of this in Barcelo 1]. Our results were announced in 10] and 13]. Some related results have since been obtained by Hanlon 8].