  We study hyperbolic polynomials with nice symmetry and express them as the
determinant of a Hermitian matrix with special structure. The goal of this
paper is to answer a question posed by Chien and Nakazato in 2015. By properly
modifying a determinantal representation construction of Dixon (1902), we show
for every hyperbolic polynomial of degree n invariant under the cyclic group of
order n there exists a determinantal representation admitted via some cyclic
weighted shift matrix. Moreover, if the polynomial is invariant under the
action of the dihedral group of order n, the associated cyclic weighted shift
matrix is unitarily equivalent to one with real entries.
