Motivated in part by this result, we show that we can expect a cyclic sieving phenomenon for m-tuples of skew standard Young tableaux of the same shape and the mth power of the associated fake-degree polynomial, for fixed m, under mild and easily checked conditions. However, we are unable to exhibit an appropriate group action explicitly.
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 N. 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 J. Stembridge's alternating tableaux, which
intertwines rotation and promotion.
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