Hostname: page-component-7479d7b7d-k7p5g Total loading time: 0 Render date: 2024-07-09T14:26:15.907Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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
Check for updates
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

References

Alexandersson, P. and Amini, N., ‘The cone of cyclic sieving phenomena’, Discrete Math. 342(6) (2019), 1581–1601. doi:10.1016/j.disc.2019.01.037.CrossRefGoogle Scholar
Alexandersson, P., Linusson, S. and Potka, S., ‘The cyclic sieving phenomenon on circular Dyck paths’, Electron. J. Combin. 26 (2019), 1–32. doi:10.37236/8720.CrossRefGoogle Scholar
Alexandersson, P. and Uhlin, J., ‘Cyclic sieving, skew Macdonald polynomials and Schur positivity’, Algebr. Comb. 3(4) (2020), 913–939. doi:10.5802/alco.123.CrossRefGoogle Scholar
Bennett, M., Madill, B. and Stokke, A., ‘Jeu-de-taquin promotion and a cyclic sieving phenomenon for semistandard hook tableaux’, Discrete Math. 319 (2014), 62–67. doi:10.1016/j.disc.2013.11.024.CrossRefGoogle Scholar
Brauer, R., ‘On the connection between the ordinary and the modular characters of groups of finite order’, Ann. of Math. (2) 42 (1941), 926–935. doi:10.2307/1968774.CrossRefGoogle Scholar
Barcelo, H., Reiner, V. and Stanton, D., ‘Bimahonian distributions’, J. Lond. Math. Soc. (2) 77(3) (2008), 627–646. doi:10.1112/jlms/jdn004.CrossRefGoogle Scholar
Désarménien, J., ‘Fonctions symétriques associées à des suites classiques de nombres’, Ann. Sci. Éc. Norm. Supér. (4) 16(2) (1983), 271–304. doi:https://doi.org/10.24033/asens.1449.CrossRefGoogle Scholar
Fontaine, B. and Kamnitzer, J., ‘Cyclic sieving, rotation, and geometric representation theory’, Selecta Math. (N.S.) 20(2) (2013), 609–625. doi:10.1007/s00029-013-0144-4.CrossRefGoogle Scholar
Fomin, S. V. and Lulov, N., ‘On the number of rim hook tableaux’, J. Math. Sci. (N.Y.) 87(6) (1997), 4118–4123. doi:10.1007/bf02355806.CrossRefGoogle Scholar
James, G. and Kerber, A., The Representation Theory of the Symmetric Group (Cambridge University Press, Cambridge, 1984). doi:10.1017/cbo9781107340732.CrossRefGoogle Scholar
Kovács, L. G., ‘The permutation lemma of Richard Brauer’, Bull. Lond. Math. Soc. 14(2) (1982), 127–128. A letter to C. W. Curtis. doi:10.1112/blms/14.2.127.CrossRefGoogle Scholar
Kuperberg, G., ‘Spiders for rank $2$ Lie algebras’, Comm. Math. Phys. 180(1) (1996), 109–151. doi:10.1007/bf02101184.CrossRefGoogle Scholar
Lascoux, A., Leclerc, B. and Thibon, J.-Y., ‘Ribbon tableaux, Hall–Littlewood functions, quantum affine algebras and unipotent varieties’, J. Math. Phys 38 (1997), 1041–1068.CrossRefGoogle Scholar
Macdonald, I. G., Symmetric Functions and Hall Polynomials, second edn, Oxford Mathematical Monographs, (The Clarendon Press, New York, 1995). With contributions by A. Zelevinsky.Google Scholar
Y.-T., Oh and Park, E., ‘$q$-dimensions of highest weight crystals and cyclic sieving phenomenon’, Preprint, 2020, arXiv:2008.03025.Google Scholar
Pak, I., ‘Ribbon tile invariants’, Trans. Amer. Math. Soc. 352(12) (2000), 5525–5562. doi:10.1090/s0002-9947-00-02666-0.CrossRefGoogle Scholar
Pfannerer, S., In preparation.Google Scholar
Pfannerer, S., Rubey, M. and Westbury, B. W., ‘Promotion on oscillating and alternating tableaux and rotation of matchings and permutations’, Algebr. Comb. 3(1) (2020), 107–141. doi:10.5802/alco.87.Google Scholar
Purbhoo, K., ‘Wronskians, cyclic group actions, and ribbon tableaux’, Trans. Amer. Math. Soc. 365(4) (2013), 1977–2030. URL: http://www.jstor.org/stable/23513431.CrossRefGoogle Scholar
Rhee, D., Cyclic Sieving Phenomenon of Promotion on Rectangular Tableaux (2012), M.S. thesis, University of Waterloo, Waterloo, Ontario, Canada. URL: https://pdfs.semanticscholar.org/c7e0/60d4db5fa5a381da333678054288ed6e97a4.pdf.Google Scholar
Rhoades, B., ‘Cyclic sieving, promotion, and representation theory’, J. Combin. Theory Ser. A 117(1) (2010), 38–76. doi:10.1016/j.jcta.2009.03.017.CrossRefGoogle Scholar
Rhoades, B., ‘Hall–Littlewood polynomials and fixed point enumeration’, Discrete Math. 310(4) (2010), 869–876. doi:10.1016/j.disc.2009.10.003.CrossRefGoogle Scholar
Robbins, H., ‘A remark on Stirling’s formula’, Amer. Math. Monthly 62(1) (1955), 26. doi:10.2307/2308012.Google Scholar
Reiner, V., Stanton, D. and White, D. E., ‘The cyclic sieving phenomenon’, J. Combin. Theory Ser. A 108(1) (2004), 17–50. doi:10.1016/j.jcta.2004.04.009.CrossRefGoogle Scholar
Rubey, M. and Westbury, B. W., ‘A combinatorial approach to classical representation theory’, Preprint, 2014, arXiv:1408.3592.Google Scholar
Springer, T. A., ‘Regular elements of finite reflection groups’, Invent. Math. 25(2) (1974), 159–198. doi:10.1007/bf01390173.CrossRefGoogle Scholar
Schilling, A., Shimozono, M. and White, D. E., ‘Branching formula for $q$-Littlewood–Richardson coefficients’, Adv. Appl. Math. 30(1-2) (2003), 258–272. doi:10.1016/s0196-8858(02)00535-3.CrossRefGoogle Scholar
Sagan, B. E., Shareshian, J. and Wachs, M. L., ‘Eulerian quasisymmetric functions and cyclic sieving’, Adv. Appl. Math. 46(1) (2011), 536–562. doi:10.1016/j.aam.2010.01.013.CrossRefGoogle Scholar
Stanley, R. P., Enumerative Combinatorics: Volume 2, first edn, (Cambridge University Press, Cambridge, 2001). doi:10.1017/CBO9780511609589.Google Scholar
Stembridge, J. R., ‘Rational tableaux and the tensor algebra of ${\mathrm{gl}}_n$’, J. Combin. Theory Ser. A 46(1) (1987), 79–120. doi:10.1016/0097-3165(87)90077-x.CrossRefGoogle Scholar
Stembridge, J. R., ‘Canonical bases and self-evacuating tableaux’, Duke Math. J. 82(3) (1996), 585–606. doi:10.1215/s0012-7094-96-08224-1.CrossRefGoogle Scholar
Westbury, B. W., ‘Invariant tensors and the cyclic sieving phenomenon’, Electron. J. Combin. 23(4) (2016), 1–40. doi:10.37236/4569.CrossRefGoogle Scholar
Westbury, B. W. ‘Interpolating between promotion and the long cycle’, Preprint, 2019, arXiv:1906.07146.Google Scholar
White, D. E., ‘A bijection proving orthogonality of the characters of ${S}_n$’, Adv. Math. 50(2) (1983), 160–186. doi:10.1016/0001-8708(83)90038-5.CrossRefGoogle Scholar