On Some Submodules of the Action of the Symmetrical Group on the Free Lie Algebra

Abstract

The free Lie algebra Lie[A] over the complex held, on an alphabet A, is the smallest subspace of the complex linear span of all words in A, which is closed under the bracket operation [u, v] = uv - vu. Define Lien to be the subspace of the free Lie algebra Lie[1, ..., n] spanned by bracketings consisting of words which are permutations of {1, ..., n}. The symmetric group Sn acts on Lien by replacement of letters, giving an (n - 1)!-dimensional representation isomorphic to the induction [omega][short up arrow]SnCn, where Cn is the cyclic group of order n and [omega] is a primitive nth root of unity. Bracketings in Lien may be represented graphically by labelled binary trees with n leaves. Fix a particular unlabelled binary tree T; then the vector subspace spanned by all words corresponding to the n! possible labellings of T is an Sn-module VT. In this paper we study the representations afforded by certain classes of trees T. We show that the plethysm VS[VT] is isomorphic to the submodule corresponding to a tree S[T] which has a natural description in terms of the trees S and T.