ON THE ACTION OF THE SYMMETRICAL GROUP ON THE FREE LIE-ALGEBRA AND THE PARTITION LATTICE

The Free Lie Algebra over an alphabet A, denoted here by LIE[A], is the smallest subspace of the linear span of the A-words which contains the letters and is closed under the bracket operation [f, g] = fg - gf. A permutation σ acts on words by replacing each occurrence of the letter ai by aσ. This action linearly extends to LIE[A]. We are concerned here with the action of the symmetric group Sn on the subspace of LIE[A] which is the linear span of bracketings of words which are permutations of the letters of the alphabet. It follows from the work of Hanlon, Stanley, and Joyal that this action and the action of Sn on the top homology of the partition lattice Πn induce similar representations (up to tensoring with the alternating character). It follows from the work of Garsia and Stanton that the action on the homology is similar to the action on a suitably defined top portion of the Stanley-Reisner ring. In this paper we derive a direct combinatorial proof of the similarity of these three actions by choosing natural bases in each of these three spaces and comparing the matrices corresponding to the simple reflections. © 1990.