Homology of the Lie algebra corresponding to a poset.

Abstract

In this thesis, 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 $\{1,2,\...,{\rm n}\},$ we define the nilpotent Lie algebra $L\sb{P}$ to be the span of all elementary matrices $z\sb{x,y}$, such that x is less than y in P. In this thesis, we will make a decisive step toward calculating the Lie algebra homology of $L\sb{P}$ in the case that the Hasse diagram of P is a rooted tree. We show that the Laplacian ${\cal L}$ is significantly simplified when the posets considered are those whose Hasse diagram is a tree. The main result of this thesis determines the spectral resolutions of three commuting linear operators whose sum is the Laplacian ${\cal L}$ of the Koszul complex of $L\sb{P}$ 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 $L\sb{P}$ is represented by the nullspace of ${\cal L}$, so in future work, these results should allow for the homology to be effectively computed. These results have several interesting corollaries that are of a combinatorial nature. We will state one. Let P be a rooted tree on n nodes and let $\Sigma$ be the sum in the group algebra of $S\sb{n}$ of all transpositions (i, j) such that i is on the unique path from j to the root in P. Then $\Sigma$ acting on C$S\sb{n}$ by left multiplication has non-negative integer eigenvalues and the corresponding eigenspaces can be identified in representation-theoretic terms.