Skew hook Schur functions and the cyclic sieving phenomenon

Abstract

Fix an integer $t \geq 2$ and a primitive $t^{\text{th}}$ root of unity $\omega$. We consider the specialized skew hook Schur polynomial $\text{hs}_{\lambda/\mu}(X,\omega X,\dots,\omega^{t-1}X/Y,\omega Y,\dots,\omega^{t-1}Y)$, where $\omega^k X=(\omega^k x_1, \dots, \omega^k x_n)$, $\omega^k Y=(\omega^k y_1, \dots, \omega^k y_m)$ for $0 \leq k \leq t-1$. We characterize the skew shapes $\lambda/\mu$ for which the polynomial vanishes and prove that the nonzero polynomial factorizes into smaller skew hook Schur polynomials. Then we give a combinatorial interpretation of $\text{hs}_{\lambda/\mu}(1,\omega^d,\dots,\omega^{d(tn-1)}/1,\omega^d,\dots,\omega^{d(tm-1)})$, for all divisors $d$ of $t$, in terms of ribbon supertableaux. Lastly, we use the combinatorial interpretation to prove the cyclic sieving phenomenon on the set of semistandard supertableaux of shape $\lambda/\mu$ for odd $t$. Using a similar proof strategy, we give a complete generalization of a result of Lee--Oh (arXiv: 2112.12394, 2021) for the cyclic sieving phenomenon on the set of skew SSYT conjectured by Alexandersson--Pfannerer--Rubey--Uhlin (Forum Math. Sigma, 2021).