A basis for the top homology of a generalized partition lattice
It is well known that partition lattices \(\Pi_n\) are Cohen-Macaulay and so the only non-trivial homology of \(\Pi_n\) occurs at the top. In 1989 \textit{H. Barcelo} [J. Comb. Theory, Ser. A 55, No. 1, 93-129 (1990; Zbl 0709.17005)] demonstrated an explicit bijection between a basis for the top homology of \(\Pi_n\) and a basis of the free Lie algebra \(\text{Lie} [a_1,\ldots,a_n]\) which preserves an action of the symmetric group \(S_n\). Note that \(\Pi_n\) may be considered as the intersection lattice defined by the hyperplane arrangement \(\{x_i-x_j=0 : i,j\in [n]\), \(i\neq j\}\). A generalized partition lattice is the intersection lattice \(L_{n,k}\) of the hyperplane arrangement \(\{x_i-x_j=m : i,j\in [n]\), \(i\neq j\), \(m\in\{0,1,\ldots,k\}\}\). By a result of \textit{M. L. Wachs} and \textit{J. W. Walker} [Order 2, No. 4, 367-385 (1986; Zbl 0589.06005)] \(L_{n,k}\) is Cohen-Macaulay. In the paper under review the author shows by computing the character of the natural action of \(S_n\) on the top homology of \(L_{n,k}\) that \(H_{n-1}(L_{n,k})\) is isomorphic as an \(S_n\)-module to a direct sum of \({\mathbb{C}}S_n\). Then an explicit basis for \(H_{n-1}(L_{n,k})\) is constructed, where the basis elements are indexed by all labelled, rooted, \((k+1)\)-ary trees on \(n\) vertices in which the root has no 0-child.
scientific article; zbMATH DE number 1275780
scientific article