The eigenvalues of the Laplacian for the homology of the Lie algebra corresponding to a poset

Abstract

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.