Weight vectors and highest weight vectors in unitary highest weight modules

abstract: The theta correspondence for the dual pair $(\widetilde{\mathbf{U}}_n,\widetilde{\mathbf{U}}_{p,q})$ (where $\widetilde{\mathbf{U}}_n$ and $\widetilde{\mathbf{U}}_{p,q}$ are 2-fold covers of $\mathbf{U}_n$ and $\mathbf{U}_{p,q}$ respectively) is used to find nice bases for a collection of unitary highest weight modules of $\widetilde{\mathbf{U}}_{p,q}$, and also for the highest weight vectors with respect to the obvious maximal compact subgroup. The techniques used involve signed Hibi rings, an extension of the concept of Hibi ring, which has been identified as an underlying structure in constructing standard monomial theories, and in finding straightening laws in certain rings. An extension of the Littlewood-Richardson rule to tensor products of arbitrary rational representations of $\mathrm{GL}_n(\mathbb{C})$ is required for finding the desired bases, and is also provided.