Skew characters and cyclic sieving

Abstract

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.

References in corpus (6)

• Cyclic sieving, skew Macdonald polynomials and Schur positivity
• The Cone of Cyclic Sieving Phenomena
• A combinatorial approach to classical representation theory
• The cyclic sieving phenomenon on circular Dyck paths
• $q$-dimensions of highest weight crystals and cyclic sieving phenomenon
• Interpolating between promotion and the long cycle

Cited by in corpus (6)

• Cyclic sieving, skew Macdonald polynomials and Schur positivity
• Promotion and cyclic sieving on families of SSYT
• $q$-dimensions of highest weight crystals and cyclic sieving phenomenon
• Free action and cyclic sieving on skew semi-standard Young tableaux
• Refined Catalan and Narayana cyclic sieving
• Tableau Stabilization and Lattice Paths