New Branching Formulae for Classical Groups and Relations among them

We find the branching laws for the classical pairs $\mathrm{GL}(m, \mathbb{C}) \subset \mathrm{GL}(n, \mathbb{C})$, $\mathrm{Sp}(2m, \mathbb{C}) \subset \mathrm{Sp}(2n, \mathbb{C})$, $\mathrm{SO}(q, \mathbb{C}) \subset \mathrm{SO}(p, \mathbb{C})$ for all $m\leq n$, and all $q\leq p$, generalizing the well-known results of classical branching laws which exist for $m=n-1$, and $q=p-1$. Our approach provides a common proof applicable to all these groups. We also compare the branching multiplicities among these pairs.