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Highest Weight Vectors in Plethysms

  • Kazufumi KimotoEmail author
  • Soo Teck Lee
Article
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Abstract

We realize the \(\mathrm {GL}_n(\mathbb {C})\)-modules \(S^k(S^m(\mathbb {C}^n))\) and \(\Lambda ^k(S^m(\mathbb {C}^n))\) as spaces of polynomial functions on \(n\times k\) matrices. In the case \(k=3\), we describe explicitly all the \(\mathrm {GL}_n(\mathbb {C})\)-highest weight vectors which occur in \(S^3(S^m(\mathbb {C}^n))\) and in \(\Lambda ^3(S^m(\mathbb {C}^n))\) respectively. In particular, we obtain alternative formulas for the multiplicities in these modules.

Notes

Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and information on several existing results on plethysms.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of the RyukyusNishiharaJapan
  2. 2.Department of MathematicsNational University of SingaporeSingaporeRepublic of Singapore

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