The spectra of Laplace operators on covering simplicial complexes

Abstract

We give a decomposition of the Laplace operator (in matrix form) of a covering simplicial complex as a direct sum of several matrices, one of which is the Laplace operator of the base complex. It follows that the spectrum of a covering simplicial complex is a multiset union of the spectrum of the base simplicial complex and the spectra of other relevant matrices, which implies the spectral inclusion property of Horak and Jost. In the case of a $2$-fold covering, we show that the spectrum is a multiset union of the spectrum of the base complex and that of an incidence-signed simplicial complex, thereby generalizing a result of Bilu and Linial from graphs to simplicial complexes. Additionally, we show that the dimension of the cohomology of a covering complex is greater than or equal to that of the base complex. Our arguments exploit the coverings of incidence graphs of simplicial complexes and the representation theory of permutation groups.