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Shellability of complexes of trees. (English) Zbl 0916.06004
Let \({\mathcal F}^{(k)}_{n}\) 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 P. Hanlon [J. Comb. Theory, Ser. A 74, 301-320 (1996; Zbl 0848.05021)] that \({\mathcal F}^{(k)}_{n}\) are Cohen-Macaulay. The paper under review provides a proof of shellability of \({\mathcal F}^{(k)}_{n}\). Additionally an explicit basis for the homology of this complex is obtained; this basis is equivalent to the basis constructed by P. Hanlon and 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.

06A11 Algebraic aspects of posets
05E20 Group actions on designs, etc. (MSC2000)
05C05 Trees
17B01 Identities, free Lie (super)algebras
Full Text: DOI
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