On the action of the symmetric group on the free Lie algebra and the partition lattice (Q921100)

Let \(A=\{a_ 1,...,a_ N\}\) and LIE[A] denote the free Lie algebra on A. Let \(LIE[a_ 1,...,a_ n]\) be a subspace of LIE[A] consisting of polynomials which are linear combinations of words \(a_{\alpha_ 1}a_{\alpha_ 2}...a_{\alpha_ n}\) which are permutations of the letters \(a_ 1,...,a_ n\). A permutation \(\sigma\) acts on words by replacing each occurence of the letter \(a_ i\) by \(a_{\sigma_ i}\). This action restricted to \(LIE[a_ 1,...,a_ n]\) induces a representation of \(S_ n\) denoted by LIEn. Let HPIn denote the representation induced by the action of \(S_ n\) on the top homology of the partition lattice \(\Pi_ n\) and SRPIn denote the representation induced by the action of \(S_ n\) on a suitably defined quotient of the Stanley-Reisner ring of the poset \(\Pi_ n\). By constructing suitable bases (instead of the character approach) for the free Lie algebra, the Stanley-Reisner ring and the homology of \(\Pi_ n\), it is shown that the representations LIEn, HPIn, and SRPIn are identical (up to tensoring and a transposition) and thus equivalent.