Decompositions of Sn-Submodules in the Free Lie Algebra

AbstractDefine Lien to be the subspace of the free Lie algebra Lie[1 ··· n] (over the complex numbers) 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 ω↑SnCn, where Cn is the cyclic group of order n and ω is a primitive nth root of unity. A sequence of bracketings can be encoded by a labelled binary tree T. The left ideal of the group algebra generated by a fixed sequence of bracketings is an Sn-module VT. In this paper we continue the study, begun in [H. Barcelo and S. Sundaram, On some sub-modules of the action of the symmetric group on the free Lie algebra, J. of Algebra, to appear] of the representations afforded by certain bracketings. We present a technique for decomposing these representations as a sum of submodules whose structure is known. In most cases, our methods allow us to calculate the decomposition into irreducibles of the representations VT, modulo a plethysm computation and applications of the Littlewood-Richardson rule. Our formulas enable us to compute the character values of these representations. In particular, we show that if n does not divide m (n < m), the induced module (Liem ⊗ Lien) ↑ is isomorphic to the subspace generated by a particular sequence of bracketings in Sm + n. We also describe classes of trees T for which the Sn-spaces VT behave nicely upon restriction to Sn − 1 we give explicit formulas to compute these restrictions