  Let $k$ and $m$ be positive integers and $\lambda/\mu$ a skew partition. We
compute the principal specialization of the skew Schur polynomials $s_{\lambda
/\mu}(x_1, \ldots, x_{k})$ modulo $q^m-1$ under suitable conditions. We
interpret the results thus obtained from the viewpoint of the cyclic sieving
phenomenon on semistandard Young skew tableaux of shape $\lambda/\mu$. As an
application, we deal with evaluations of the principal specialization of the
skew Schur polynomials at roots of unity.
