Highest Weight Vectors in Plethysms
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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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