Skew characters and cyclic sieving

Abstract

In 2010, B. Rhoades proved that promotion together with the fakedegree 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 fakedegree 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.

Promotion and cyclic sieving on families of SSYT

“…Finally, a different approach would be to keep the Kostka-Foulkes polynomials, and replace promotion with a different group action. In [APRU20], it is shown that the set of standard Young tableaux of shape mλ := (mλ 1 , mλ 2 , . .…”

“…It can be shown (see e.g., [APRU20]) that ε k (λ/µ) can be given in terms of a skew character, but we do not need that here.…”

“…In a recent preprint, [APRU20], it is proved that for a fixed skew shape λ/µ, there exist some cyclic group action C n of order n, such that SYT(nλ/nµ), C n , f nλ/nµ (q)…”

Refined Catalan and Narayana cyclic sieving

“…We discuss an open problem regarding the interpolation between type A and type B q-Narayana numbers. This problem is part of a broader set of questions regarding the interplay of cyclic sieving and characters in the symmetric group, see [APRU20]. We argue that the small special case discussed below is interesting in its own right.…”

$q$-dimensions of highest weight crystals and cyclic sieving phenomenon

“…, m}. The cyclic sieving phenomenon about SST m (λ) and pr has drawn big attention from many researchers (see [2,3,4,5,8,21,22,29] for example.) One of the most celebrated researches in this direction was initiated by Rhoades [21].…”