Skew hook Schur functions and the cyclic sieving phenomenon

TL;DR

This paper investigates specialized skew hook Schur functions at roots of unity, characterizes when they vanish, and demonstrates their connection to the cyclic sieving phenomenon through combinatorial interpretations involving ribbon supertableaux.

Contribution

It provides a characterization of when these specialized skew hook Schur functions are nonzero, factorizes them into smaller components, and establishes their role in cyclic sieving phenomena using combinatorial models.

Findings

Characterization of vanishing skew hook Schur functions

Factorization into smaller skew hook Schur polynomials

Cyclic sieving phenomenon on supertableaux

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).