The eigenvalues of the Laplacian for the homology of the Lie algebra corresponding to a poset
type: Item
instance of: scholarly article
title: The eigenvalues of the Laplacian for the homology of the Lie algebra corresponding to a poset
zbMATH Open document ID: 0827.58059
author: Iztok Hozo
published in: The Electronic Journal of Combinatorics
publication date: 1995-09-11T00:00:00Z
full work available at URL: https://eudml.org/doc/118748
full work available at URL: R14
review text: 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.
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 782546
zbMATH Keywords: spectral resolution
zbMATH Keywords: Laplacian
zbMATH Keywords: Koszul complex
zbMATH Keywords: posets
zbMATH Keywords: Lie algebra
zbMATH Keywords: Hasse diagram
zbMATH Keywords: eigenvalues
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 782546