Skew Pieri algebras of the general linear group

Let V be an irreducible polynomial representation of the general linear group GLn=GLn(C) and let α₁, ⋯, α_q be nonnegative integers less than or equal to n. We call a description of the irreducible decomposition of the tensor product V⊕ - Λα1(Cn) ⊕ - Λαq(Cn) an iterated skew Pieri rule for GL_n. In this paper, we define a family of complex algebras whose structure encodes an iterated skew Pieri rule for GL_n, and we call these algebras iterated skew Pieri algebras. Our main goal is to construct a basis for each of these algebras thereby giving explicit highest weight vectors in the above tensor product.