Some surprising similarities between matroids and shifted simplicial
Abstract: Matroid complexes and shifted complexes are two of only a handful of families of simplicial complexes whose combinatorial Laplacian eigenvalues are known to be all integers. Kook found a recursion relation expressing the Laplacian eigenvalues of a matroid complex 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 relative complex (M\e, M/e). We further show that by generalizing the contraction of a matroid complex to the link of an arbitrary simplicial complex, the Laplacian eigenvalues of shifted complexes satisfy this exact same recursion.