A common recursion for Laplacians of matroids and shifted simplicial complexes

Abstract

A recursion due to Kook expresses the Laplacian eigenvalues of a matroid $M$ in terms of the eigenvalues of its deletion $M-e$ and contraction $M/e$ by a fixed element $e$, and an error term. We show that this error term is given simply by the Laplacian eigenvalues of the pair $(M-e,M/e)$. We further show that by suitably generalizing deletion and contraction to arbitrary simplicial complexes, the Laplacian eigenvalues of shifted simplicial complexes satisfy this exact same recursion. We show that the class of simplicial complexes satisfying this recursion is closed under a wide variety of natural operations, and that several specializations of this recursion reduce to basic recursions for natural invariants. We also find a simple formula for the Laplacian eigenvalues of an arbitrary pair of shifted complexes in terms of a kind of generalized degree sequence.