In 2010, Rhoades proved that promotion on rectangular standard Young tableaux, together with the associated fake-degree polynomial, provides an instance of the cyclic sieving phenomenon. We extend this result to m-tuples of skew standard Young tableaux of the same shape, for fixed m, subject to the condition that the mth power of the associated fake-degree polynomial evaluates to nonnegative integers at roots of unity. However, we are unable to specify an explicit group action. Put differently, we determine in which cases the mth tensor power of a skew character of the symmetric group carries a permutation representation of the cyclic group. To do so, we use a method proposed by Amini and the first author, which amounts to establishing a bound on the number of border-strip tableaux of skew shape. Finally, we apply our results to the invariant theory of tensor powers of the adjoint representation of the general linear group. In particular, we prove the existence of a bijection between permutations and Stembridge’s alternating tableaux, which intertwines rotation and promotion.

中文翻译：

---

倾斜字符和循环筛分

2010 年，Rhoades 证明了在矩形标准 Young 画面上的提升以及相关的伪度多项式提供了循环筛分现象的一个实例。我们将此结果扩展到米-相同形状的倾斜标准Young tableaux的元组，用于固定米, 条件是米相关联的伪度多项式的 th 次方计算为单位根处的非负整数。但是，我们无法指定明确的组操作。换句话说，我们确定在哪些情况下米对称群的偏斜特征的张量幂携带循环群的置换表示。为此，我们使用了 Amini 和第一作者提出的方法，该方法相当于对倾斜形状的边界条画面的数量建立了界限。最后，我们将我们的结果应用于一般线性群的伴随表示的张量幂的不变理论。特别是，我们证明了排列和 Stembridge 的交替画面之间存在双射，它将旋转和提升交织在一起。