A proof of Hozo's conjecture on the homology of a class of nilpotent Lie algebras

[摘要] Let P be a rooted tree with root r and vertex set {1, 2, ..., n}. Consider P to be a poset by saying u less than or equal to v if u is on the path from r to v. Let Z[P] be the span of all matrices z(uv) such that u <(P) v, where z(uv) is the n X n matrix with a 1 in the u, v entry and 0's elsewhere. Note that Z[P] is a nilpotent Lie subalgebra of sl(n)(C). I. Hozo (Electron. J. Combin., to appear) studied the Laplacian of the Koszul complex for computing the Lie algebra homology of Z[P]. He showed that the tree structure of P forces significant simplifications in the Laplacian. He went on to conjecture that the eigenvalues of the Laplacian are indexed by certain labellings of P by partitions. He gave a simple method for determining the eigenvalue indexed by a labelling. In this paper we prove Hozo's conjecture. As a consequence, we deduce that all eigenvalues of the Laplacian are integers. (C) 1997 Academic Press.

[发布日期] 1997-04-15