Bases for some reciprocity algebras I

For a complex vector space $V$, let $\mathcal{P}(V)$ be the algebra of polynomial functions on $V$. In this paper, we construct bases for the algebra of all $\operatorname{GL}_n(\mathbb{C})\times \operatorname{GL}_{m_1} (\mathbb{C})\tim... ..._2} (\mathbb{C})\times\cdot\cdot\cdot\times \operatorname{GL}_{m_r}(\mathbb{C})$ highest weight vectors in $\mathcal{P}\left(\mathbb{C}^n\otimes \mathbb{C}^m\right)$, where $m=m_1+\cdot\cdot\cdot+m_r$ and $m_j\leq n$ for all $1\leq j\leq r$, and the algebra of $\operatorname{GL}_n(\mathbb{C})\times \operatorname{GL}_k(\mathbb{C})\times\operatorname{GL}_1(\mathbb{C})$ highest weight vectors in $\mathcal{P}\left[\left(\mathbb{C}^n\otimes\mathbb{C}^k\right)\oplus \left(\mathbb{C}^n{}^\ast\otimes\mathbb{C}^l\right)\right]$.