  Here we study the space of real hyperbolic plane curves that are invariant
under actions of the cyclic and dihedral groups and show they have
determinantal representations that certify this invariance. We show an analogue
of Nuij's theorem for the set of invariant hyperbolic polynomials of a given
degree. The main theorem is that every invariant hyperbolic plane curve has a
determinantal representation using a block cyclic weighted shift matrix. This
generalizes previous work by Lentzos and the first author, as well as by Chien
and Nakazato. One consequence is that if the numerical range of a matrix is
invariant under rotation, then it is the numerical range of a block cyclic
weighted shift matrix.
