We study the subspace of the exterior algebra of a simple complex Lie algebra linearly spanned by the copies of the little adjoint representation or, in the case of the Lie algebra of traceless matrices, by the copies of the -th symmetric power of the defining representation. As main result we prove that this subspace is a free module over the subalgebra of the exterior algebra generated by all primitive invariants except the one of highest degree.
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Corrado De Concini, Pierluigi Möseneder Frajria, Paolo Papi, Claudio Procesi, On special covariants in the exterior algebra of a simple Lie algebra. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 25 (2014), no. 3, pp. 331–344