This is the second part of a paper, where, unfortunately, the first part, from which notations and results are quoted, hasn’t appeared yet. Several results come from the author’s thesis [Universal operations in the representation theory of groups (Thesis, Utrecht 1986)]. The joint study of representations of symmetric and polynomial representations of general linear groups goes back to I. Schur and was heavily influenced by J. A. Green’s lecture notes [Polynomial representations of \(GL_ n\) (Lect. Notes Math. 830, 1980; Zbl 0451.20037)] and his introduction of the Schur algebra.
On the other hand, M. Clausen has used a polynomial ring in doubly indexed indeterminates to model the various module (Specht and Weyl) constructions [in Adv. Math. 33, 161-191 (1979; Zbl 0425.20011); see also J. Symb. Comput. 11, 483-522 (1991) for a more recent treatment].
The author continues this program to study further connections between the Weyl module and its contravariant dual, the Schur module, by means of homomorphisms between them and natural transformations between the Weyl and Schur functors. In addition to the method of letter place algebras a generalization of Steinberg’s tensor product theorem is of importance.