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\((\mathrm{GL}_n,\mathrm{GL}_m)\)-duality and symmetric plethysm. (English) Zbl 0705.20040

Summary: In [Proc. Symp. Pure Math. 48, 133–166 (1988; Zbl 0661.15028)] the author has given an exposition of the theory of invariants of binary forms in terms of a particular version of Classical invariant Theory. Reflection shows that many aspects of the development apply also to \(n\)-ary forms. The purpose of this paper is to make explicit this more general application. The plethysms \(S^{\ell}(S^ p(\mathbb C^ n))\) are computed quite explicitly for \(\ell =2,3\) and \(4\).

MSC:

20G05 Representation theory for linear algebraic groups
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
15A72 Vector and tensor algebra, theory of invariants
15A69 Multilinear algebra, tensor calculus
14L30 Group actions on varieties or schemes (quotients)
22E10 General properties and structure of complex Lie groups

Citations:

Zbl 0661.15028
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References:

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