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

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, . . . , n}, we define the nilpotent Lie algebra LP to be the span of all elementary matrices zx,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 LP in the case that the Hasse diagram of P is a rooted tree. We show that the Laplacian 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 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 Classification Number: 17B56 (primary) 05E25 (secondary) 1 Preliminaries 1.1 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 ∈ 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 chain if C is a chain when regarded as a subposet of P . Definition 1.1 A poset P is linear if for any two comparable elements x, y ∈ P , the interval [x, y] is a chain, i.e., if every interval has the structure of a chain. The length l(C) of a finite chain C is defined by l(C) = |C| − 1. 1.2 The homology of a poset The combinatorial approach to a homology theory for posets was developed by Rota [29], Farmer [8], Lakser [22], Mather [25], Crapo [5] and others (more references can be found in [33]). A systematic development of the relationship between the combinatorial and topological properties of posets was begun by K. Baclawski [1] and A. Björner [2] and continued by J. Walker [33]. Define the set Cr(P ) to be the set of 0-1 chains of length r in the poset P . By abuse of notation we will use the same name for the complex vector space Cr or Cr(P ), with basis the set of r-chains. The Cr’s are called chain spaces. The map ∂r : Cr → Cr−1, called the boundary map, is defined by: ∂r(0̂ < x1 < . . . < xr < 1̂) = r ∑ i=1 (−1)i−1(0̂ < x1 < . . . < x̂i < . . . < xr < 1̂) It is easy to check that: Lemma 1 ∂r−1 ◦ ∂r = 0. This allows us now to define the homology of a poset to be: Hr(P ) = Ker(∂r)/Im(∂r+1) Later in this work we will talk about an operator, called the Laplacian of a complex, for which we need to identify the transpose of the boundary map. We are in fact transposing the matrix of the boundary map with respect to the basis of r-chains. In this case the case of the poset homology, the transpose of the boundary map is not so difficult to evaluate. Lemma 2 The transpose of the boundary operator (viewed as a linear map), is given by the following expression: ∂(0̂ < x1 < . . . < xr < 1̂) = r ∑