  Each degree $n$ polynomial in one variable of the form
$(x+1)(x^{n-1}+c_1x^{n-2}+\cdots +c_{n-1})$ is representable in a unique way as
a Schur-Szeg\H{o} composition of $n-1$ polynomials of the form
$(x+1)^{n-1}(x+a_i)$, see \cite{Ko1}, \cite{AlKo} and \cite{Ko2}. Set $\sigma
_j:=\sum _{1\leq i_1<\cdots <i_j\leq n-1}a_{i_1}\cdots a_{i_j}$. The
eigenvalues of the affine mapping $(c_1,\ldots ,c_{n-1})\mapsto (\sigma
_1,\ldots ,\sigma _{n-1})$ are positive rational numbers and its eigenvectors
are defined by hyperbolic polynomials (i.e. with real roots only). In the
present paper we prove interlacing properties of the roots of these
polynomials.
