Highest Weight Vectors in Plethysms

  • Kazufumi KimotoEmail author
  • Soo Teck Lee


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.



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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© 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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