The eigenvalues of the Laplacian for the homology of the Lie algebra corresponding to a poset
Summary: We study the spectral resolution of the Laplacian \(\mathcal L\) of the Koszul complex of the Lie algebras corresponding to a certain class of posets. Given a poset \(P\) on the set \(\{1,2, \dots, n\}\), we define the nilpotent Lie algebra \(L_P\) to be the span of all elementary matrices \(z_{x,y}\), such that \(x\) is less than \(y\) in \(P\). In this paper, we make a decisive step toward calculating the Lie algebra homology of \(L_P\) in the case that the Hasse diagram of \(P\) is a rooted tree. We show that the Laplacian \(\mathcal L\) simplifies significantly when the Lie algebra corresponds to a poset whose Hasse diagram is a tree. The main result of this paper determines the spectral resolutions of three commuting linear operators whose sum is the Laplacian \(\mathcal L\) of the Koszul complex of \(L_P\) in the case that the Hasse diagram is a rooted tree. We show that these eigenvalues are integers, give a combinatorial indexing of these eigenvalues and describe the corresponding eigenspaces in representation-theoretic terms. The homology of \(L_P\) is represented by the nullspace of \(\mathcal L\), so in future work, these results should allow for the homology to be effectively computed.
scientific article; zbMATH DE number 782546