论文标题
偏斜钩schur函数和环状筛分现象
Skew hook Schur functions and the cyclic sieving phenomenon
论文作者
论文摘要
修复整数$ t \ geq 2 $和原始$ t^{\ text {th}} $ Unity $ω$的根。我们考虑专业的偏度钩schur schur多项式$ \ text {hs} _ {λ/μ}（x，x，ωx，\ dots，ω^{t-1} x/y，x/y，ωy，\ dots，ω^{t-1} y） $ω^k y =（ω^k y_1，\ dots，ω^k y_m）$ $ 0 \ leq k \ leq t-1 $。我们表征了偏斜的形状$λ/μ$，多项式消失并证明非零多项式分解为较小的偏斜钩schur多项式。 Then we give a combinatorial interpretation of $\text{hs}_{λ/μ}(1,ω^d,\dots,ω^{d(tn-1)}/1,ω^d,\dots,ω^{d(tm-1)})$, for all divisors $d$ of $t$, in terms of ribbon supertableaux.最后，我们使用组合解释来证明在Shape $λ/μ$的Semistandard Supertableaux上的循环筛分现象。使用类似的证明策略，我们将Lee-Oh（Arxiv：2112.12394，2021）的结果完全概括，用于循环筛分现象，该现象是由Alexandersson-Pfannerer-pfannerer--pfannerer--rubey--rubey-rubey-ububey-- rubey-- rubey-- rubey--rubey-uhlin-uhlin-uhlin（forum .uhlin（forum ohhlin）（Foram .uhlin（Foram .sigma，20211））。
Fix an integer $t \geq 2$ and a primitive $t^{\text{th}}$ root of unity $ω$. We consider the specialized skew hook Schur polynomial $\text{hs}_{λ/μ}(X,ωX,\dots,ω^{t-1}X/Y,ωY,\dots,ω^{t-1}Y)$, where $ω^k X=(ω^k x_1, \dots, ω^k x_n)$, $ω^k Y=(ω^k y_1, \dots, ω^k y_m)$ for $0 \leq k \leq t-1$. We characterize the skew shapes $λ/μ$ 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 $\text{hs}_{λ/μ}(1,ω^d,\dots,ω^{d(tn-1)}/1,ω^d,\dots,ω^{d(tm-1)})$, 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 $λ/μ$ 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).
