Double Pieri algebras and iterated Pieri algebras for the classical groups

We study iterated Pieri rules for representations of classical groups. That is, we consider tensor products of a general representation with multiple factors of representations corresponding to one-rowed Young diagrams (or in the case of the general linear group, also the duals of these). We define {\it iterated Pieri algebras}, whose structure encodes the irreducible decompositions of such tensor products. We show that there is a single family of algebras, which we call {\it double Pieri algebras}, and which can be used to describe the iterated Pieri algebras for all three families of classical groups. Furthermore, we show that the double Pieri algebras have flat deformations to Hibi rings on explicitly described posets. As an interesting application, we describe the branching rules for certain unitary highest weight modules.