Highest Weight Vectors in Plethysms

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.

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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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Correspondence to Kazufumi Kimoto.

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Kazufumi Kimoto is partially supported by Grant-in-Aid for Scientific Research (C) No. 18K03248, JSPS and by JST CREST Grant Number JPMJCR14D6, Japan. Soo Teck Lee is supported by NUS Grant R-146-000-252-114.

Communicated by Y. Kawahigashi

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Kimoto, K., Lee, S.T. Highest Weight Vectors in Plethysms. Commun. Math. Phys. 378, 1817–1841 (2020). https://doi.org/10.1007/s00220-019-03639-6

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