Bases for some reciprocity algebras I

Abstract

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 $\operatorname {GL}_n(\mathbb {C})\times \operatorname {GL}_{m_1} (\mathbb {C})\times \operatorname {GL}_{m_2} (\mathbb {C})\times \cdot \cdot \cdot \times \operatorname {GL}_{m_r}(\mathbb {C})$ 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 $\mathcal {P}\left [\left (\mathbb {C}^n\otimes \mathbb {C}^k\right )\oplus \left (\mathbb {C}^n{}^\ast \otimes \mathbb {C}^l\right )\right ]$.