Skew Schur polynomials and cyclic sieving phenomenon

TL;DR

This paper computes principal specializations of skew Schur polynomials modulo roots of unity and interprets these results through the cyclic sieving phenomenon on skew tableaux, providing new insights into their algebraic and combinatorial properties.

Contribution

It introduces a method to evaluate skew Schur polynomials at roots of unity and connects these evaluations to the cyclic sieving phenomenon on skew tableaux.

Findings

Principal specialization modulo $q^m-1$ computed under certain conditions

Interpretation of results via cyclic sieving phenomenon

Evaluations at roots of unity explored

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.