| Abstract |
Let ${\cal F}^{(k)}{n}$ be a simplicial complex of dimension $n-2$ whose facets correspond to the leaf-labelled trees with $n$ interior vertices of degree exactly $k+1$. This complex has interesting applications in homotopy theory, in the representation theory of symmetric groups and in the theory of free Lie $k$-algebras. Previously it has been proved by {\it P. Hanlon} [J. Comb. Theory, Ser. A 74, 301-320 (1996; Zbl 0848.05021)] that ${\cal F}^{(k)}{n}$ are Cohen-Macaulay. The paper under review provides a proof of shellability of ${\cal F}^{(k)}_{n}$. Additionally an explicit basis for the homology of this complex is obtained; this basis is equivalent to the basis constructed by {\it P. Hanlon} and {\it M. Wachs} [Adv. Math. 113, 206-236 (1995; Zbl 0844.17001)] for the multiplicity-free part of the free Lie $k$-algebra. The main result of this paper was also obtained independently by M. Wachs. |
