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Cyclic sieving and orbit harmonics

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Abstract

Orbit harmonics is a tool in combinatorial representation theory which promotes the (ungraded) action of a linear group G on a finite set X to a graded action of G on a polynomial ring quotient by viewing X as a G-stable point locus in \({\mathbb {C}}^n\). The cyclic sieving phenomenon is a notion in enumerative combinatorics which encapsulates the fixed-point structure of the action of a finite cyclic group C on a finite set X in terms of root-of-unity evaluations of an auxiliary polynomial X(q). We apply orbit harmonics to prove cyclic sieving results.

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Notes

  1. up to duality, but permutation representations are self-dual

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Acknowledgements

J. Oh was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2020R1A6A3A13076804). B. Rhoades was partially supported by NSF Grant DMS-1500838 and DMS-1953781. The authors are grateful to Soichi Okada, Vic Reiner, and Josh Swanson for helpful conversations. The authors also thank referees for careful reading and helpful comments.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Seoul National University, Seoul, 08826, South Korea

    Jaeseong Oh

  2. Department of Mathematics, University of California, San Diego, 9500 Gilman Dr., La Jolla, CA, 92093, USA

    Brendon Rhoades

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  1. Jaeseong Oh
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  2. Brendon Rhoades
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Correspondence to Jaeseong Oh.

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Oh, J., Rhoades, B. Cyclic sieving and orbit harmonics. Math. Z. 300, 639–660 (2022). https://doi.org/10.1007/s00209-021-02800-z

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