Skew hook Schur functions and the cyclic sieving phenomenon

Abstract

Fix an integer t2t \geq 2 and a primitive ttht^{\text{th}} root of unity ω\omega. We consider the specialized skew hook Schur polynomial hsλ/μ(X,ωX,,ωt1X/Y,ωY,,ωt1Y)\text{hs}_{\lambda/\mu}(X,\omega X,\dots,\omega^{t-1}X/Y,\omega Y,\dots,\omega^{t-1}Y), where ωkX=(ωkx1,,ωkxn)\omega^k X=(\omega^k x_1, \dots, \omega^k x_n), ωkY=(ωky1,,ωkym)\omega^k Y=(\omega^k y_1, \dots, \omega^k y_m) for 0kt10 \leq k \leq t-1. We characterize the skew shapes λ/μ\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 hsλ/μ(1,ωd,,ωd(tn1)/1,ωd,,ωd(tm1))\text{hs}_{\lambda/\mu}(1,\omega^d,\dots,\omega^{d(tn-1)}/1,\omega^d,\dots,\omega^{d(tm-1)}), for all divisors dd of tt, 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 λ/μ\lambda/\mu for odd tt. 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).Comment: 13 pages, 2 figure

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