Skew characters and cyclic sieving

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...

Full description

Saved in:
Bibliographic Details
Published in:Forum of mathematics. Sigma 2021, Vol.9, p.1, Article e41
Main Authors: Alexandersson, Per, Pfannerer, Stephan, Rubey, Martin, Uhlin, Joakim
Format: Article
Language:English
Subjects:
05E05
05E10
Discrete Mathematics
Group theory
Mathematical analysis
Permutations
Polynomials
Representations
Tensors
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:2050-5094
2050-5094
DOI:10.1017/fms.2021.11