Skew Schur polynomials and cyclic sieving phenomenon

Abstract

Let $k$ and $m$ be positive integers and $\lambda /\mu$ a skew partition. We compute the principal specialization of the skew Schur polynomials $s_{\lambda /\mu }(x_1,\dots ,x_k)$ 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.

22 pages; v2: section 5 is added, section 3 is revised, minor typos are revised; v3: Remark (2) is added in section 6, minor typos are revised; to be published in E-JC