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Twisted commutative algebras (tca's) have played an important role in thenascent field of representation stability. Let A_d be the complex tca freelygenerated by d indeterminates of degree 1. In a previous paper, we determinedthe structure of the category of A_1-modules (which is equivalent to thecategory of FI-modules). In this paper, we establish analogous results for thecategory of A_d-modules, for any d. Modules over A_d are closely related to thestructures used by the authors in previous works studying syzygies of Segre andVeronese embeddings, and we hope the results of this paper will eventually leadto improvements on those works. Our results also have implications inasymptotic commutative algebra.

GL-equivariant modules over polynomial rings in infinitely many variables. II