{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:24Z","timestamp":1753893804468,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>In this paper  we study the spectral resolution of the Laplacian ${\\cal L}$  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 ${\\cal L}$ 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 ${\\cal L}$ 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 ${\\cal L}$, so in future work, these  results should allow for the homology to be effectively computed.<\/jats:p>","DOI":"10.37236\/1208","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T01:25:54Z","timestamp":1578705954000},"source":"Crossref","is-referenced-by-count":1,"title":["The eigenvalues of the Laplacian for the homology of the  Lie algebra corresponding to a poset"],"prefix":"10.37236","volume":"2","author":[{"given":"Iztok","family":"Hozo","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[1995,7,21]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v2i1r14\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v2i1r14\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T09:31:28Z","timestamp":1579339888000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v2i1r14"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,7,21]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[1995,1,1]]}},"URL":"https:\/\/doi.org\/10.37236\/1208","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[1995,7,21]]},"article-number":"R14"}}