  A characteristic matrix function captures the spectral information of a
bounded linear operator in a matrix-valued function. In this article, we
consider a delay differential equation with one discrete time delay and assume
this equation is equivariant with respect to a compact symmetry group. Under
this assumption, the delay differential equation can have discrete wave
solutions, i.e. periodic solutions that have a discrete group of
spatio-temporal symmetries. We show that if a discrete wave solution has a
period that is rationally related to the time delay, then we can determine its
stability using a characteristic matrix function. The proof relies on
equivariant Floquet theory and results by Kaashoek and Verduyn Lunel on
characteristic matrix functions for classes of compact operators. We discuss
applications of our result in the context of delayed feedback stabilization of
periodic orbits.
