  In 2010, B. Rhoades proved that promotion together with the fake-degree
polynomial associated with rectangular standard Young tableaux give an instance
of the cyclic sieving phenomenon.
  We extend this result to all skew standard Young tableaux where the
fake-degree polynomial evaluates to nonnegative integers at roots of unity,
albeit without being able to specify an explicit group action. Put differently,
we determine in which cases a skew character of the symmetric group carries a
permutation representation of the cyclic group.
  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.
