A Relative Laplacian Spectral Recursion

The Laplacian spectral recursion, satisfied by matroid complexes and shifted complexes, expresses the eigenvalues of the combinatorial Laplacian of a simplicial complex in terms of its deletion and contraction with respect to vertex e, and the relative simplicial pair of the deletion modulo the contraction. We generalize this recursion to relative simplicial pairs, which we interpret as intervals in the Boolean algebra. The deletion modulo contraction term is replaced by the result of removing from the interval all pairs of faces in the interval that differ only by vertex e. We show that shifted pairs and some matroid pairs satisfy this recursion. We also show that the class of intervals satisfying this recursion is closed under a wide variety of operations, including duality and taking skeleta.