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Skew characters and cyclic sieving

Part of: Algebraic combinatorics

Published online by Cambridge University Press:  21 May 2021

Per Alexandersson Open the ORCID record for Per Alexandersson [Opens in a new window] ,
Stephan Pfannerer ,
Martin Rubey  and
Joakim Uhlin
Per Alexandersson
Affiliation:
Department of Mathematics, Stockholm University, S-10691, Stockholm, Sweden; E-mail: per.w.alexandersson@gmail.com
Stephan Pfannerer
Affiliation:
Fakultät für Mathematik und Geoinformation, TU Wien, Austria; E-mail: stephan.pfannerer@tuwien.ac.at, martin.rubey@tuwien.ac.at
Martin Rubey
Affiliation:
Fakultät für Mathematik und Geoinformation, TU Wien, Austria; E-mail: stephan.pfannerer@tuwien.ac.at, martin.rubey@tuwien.ac.at
Joakim Uhlin
Affiliation:
Department of Mathematics, Stockholm University, S-10691, Stockholm, Sweden; E-mail: joakim_uhlin@hotmail.com

Abstract

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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.

MSC classification

Primary: 05E10: Combinatorial aspects of representation theory
Secondary: 05E05: Symmetric functions and generalizations
Type
Discrete Mathematics
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e41
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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