[Part I, cf. Commun. Algebra 18, 4047-4086 (1990; Zbl 0732.22019); Part II, cf. ibid. 20, 2903-2917 (1992; Zbl 0780.22008).]
The authors continue their investigation of tensor operators transforming under finite dimensional representations of semisimple Lie groups. Let \(E\) be an \(n\)-dimensional vector space over the complex field \(\mathbb{C}\). The \(\mathbb{C}\)-algebra constructed in a basis free way from \(E\) is called the “shape algebra” \(\Lambda^ +E\). It is the direct sum of precisely one copy of each finite dimensional irreducible polynomial representation \(\Lambda^ \alpha E\) of the general linear group \(\text{GL}(E)\), where \(\alpha=(m_ 1,\dots,m_ s)\), \(m_ 1\geq m_ 2\geq\dots \geq m_ s\geq 1\). The authors use the method of undetermined coefficients to investigate natural transformations \(B: \Lambda^ \alpha\to \Lambda^{m_ 1}\otimes \dots\otimes \Lambda^{m_ s}\) and \(B_ t: \Lambda^ \alpha\to \Lambda^{(m_ 1,\dots, \widehat{m}_ t,\dots, m_ s)} \otimes \Lambda^{m_ t}\), where \(1\leq t\leq s\). Necessary and sufficient conditions were found on the coefficients defining \(B\) and sufficient conditions on the coefficients defining \(B_ t\). The authors also give explicit formulas for the transformations \(B_ 1\) and \(B_ s\).