Laplacians on Shifted Multicomplexes

The Laplacian of an undirected graph is a square matrix, whose eigenvalues yield important information. We can regard graphs as one-dimensional simplicial complexes, and as whether there is a generalisation of the Laplacian operator to simplicial complexes. It turns out that there is, and that is useful for calculating real Betti numbers [8]. Duval and Reiner [5] have studied Laplacians of a special class of simplicial complexes, the so called shifted simplicial complexes. They show that such Laplacians have integral spectra, computeable by a simple combinatorial formula. Snellman [11] studied a family of monomial algebras, corresponding to truncations of the ring of arithmetical functions with Dirichlet convolution. The defining monomial ideals turn out to be strongly stable, which makes it possible to use the so-called Eliahou-Kervaire resolution [7] to calculate the Betti numbers of the ideals. Now, simplicial ideals correspond to square-free monomial ideals, or differently put, to monomial ideals in the exterior alegbra. Shifted simplicial complexes correspond to strongly stable square-free ideals. Monomial ideals, on the other hand, correspond to so-called multicomplexes. One is naturally led to the question: is there a way to define Laplacian operators on finite multicomplexes, and if so, is there a simple formula for their spectra in the case of multicomplexes corresponding to strongly stable monomial ideals? The Laplacian operator on simplicial complexes is defined as Ld = ∂d+1∂ ∗ d+1, (1)