Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: The eigenvalues of the Laplacian for the homology of the Lie algebra corresponding to a poset. | The eigenvalues of the Laplacian for the homology of the Lie algebra corresponding to a poset Iztok Hozo Department of Mathematics Indiana University Northwest Gary In 46408 email ihozo@ Submitted April 6 1995 Accepted July 21 1995. Abstract In this paper we study the spectral resolution of the Laplacian 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 . ng we dehne the nilpotent Lie algebra Lp to be the span of all elementary matrices zxy such that x is less than y in P. In this paper we make a decisive step toward calculating the Lie algebra homology of Lp in the case that the Hasse diagram of P is a rooted tree. We show that the Laplacian L simplihes signihcantly 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 L of the Koszul complex of Lp 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 Lp is represented by the nullspace of L so in future work these results should allow for the homology to be effectively computed. AMS Classihcation Number 17B56 primary 05E25 secondary 1 Preliminaries Definitions A partially ordered set P or poset for short is a set which by abuse of notation we also call P together with a binary relation denoted or p when there is a possibility of confusion satisfying the following three axioms 1. For all x 2 P x x. reflexivity 2. If x y and y x then x y. antisymmetry 3. If x y and y z then x z. transitivity THE ELECTRONIC .JOURNAL OF COMBINATORICS 2 1995 R14 2 A chain or totally ordered set or linearly ordered set is a poset in which any two elements are comparable. A subset C of a poset P is called a .
