In my talk, I will present how concepts borrowed from group theory can enable a new paradigm in the study of wave propagation in layered media. The talk will consist of a review of the algebraic properties of the special linear groups and how it naturally emerges from the study of transfer matrices. Then, I will discuss some recent applications in phononics of a particular result derived thanks to this new approach, namely, the harmonic decomposition of the half-trace function (which in turn controls the dispersion relation in the medium), and how it can be applied to device metafoundations.