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Plethysm and Lattice Point Counting

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Abstract

We apply lattice point counting methods to compute the multiplicities in the plethysm of \(\textit{GL}(n)\). Our approach gives insight into the asymptotic growth of the plethysm and makes the problem amenable to computer algebra. We prove an old conjecture of Howe on the leading term of plethysm. For any partition \(\mu \) of 3, 4, or 5, we obtain an explicit formula in \(\lambda \) and k for the multiplicity of \(S^\lambda \) in \(S^\mu (S^k)\).

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Acknowledgments

The authors would like to thank Sven Verdoolaege for his prompt responses to issues raised on the isl -mailing list. The second author would like to thank Laurent Manivel for introducing him to the subject of plethysm. After the first posting of this paper on the arXiv Matthias Christandl, Laurent Manivel, and Michèle Vergne provided very insightful comments on how to apply Meinrenken-Sjamaar theory to the plethysm. We would like to thank them also for their suggestions on how to improve the paper. This project started, while Michałek was an Oberwolfach Leibniz fellow and invited Kahle for work at MFO. The project was finished at Freie Universität Berlin during Michałek’s DAAD PRIME fellowship.

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Authors and Affiliations

  1. Fakultät für Mathematik, Otto-von-Guericke Universität, 39106, Magdeburg, Germany

    Thomas Kahle

  2. Polish Academy of Sciences, Śniadeckich 8, 00-656, Warsaw, Poland

    Mateusz Michałek

  3. Simons Institute for the Theory of Computing, UC Berkeley, 121 Calvin Lab 2190, Berkeley, CA, 94720, USA

    Mateusz Michałek

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  1. Thomas Kahle
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  2. Mateusz Michałek
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Correspondence to Thomas Kahle.

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Communicated by Peter Bürgisser.

Mateusz Michałek is supported by Polish National Science Center Grant No. 2012/05/D/ST1/01063.

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Kahle, T., Michałek, M. Plethysm and Lattice Point Counting. Found Comput Math 16, 1241–1261 (2016). https://doi.org/10.1007/s10208-015-9275-7

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