For s_λ(μ_t, z, z^{-1}), the evaluation is zero or a signed product of three hyperbolic sine factors read from the t-residue profile, for every t and every shape.
Skew Schur polynomials and cyclic sieving phenomenon
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abstract
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.
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Factorization of Schur polynomials twisted by roots of unity and a reciprocal pair
For s_λ(μ_t, z, z^{-1}), the evaluation is zero or a signed product of three hyperbolic sine factors read from the t-residue profile, for every t and every shape.