New Branching Formulae for Classical Groups and Relations among them

We find the branching laws for the classical pairs $\mathrm{G}\mathrm{L}(m,\mathbb{C})\subset \mathrm{G}\mathrm{L}(n,\mathbb{C})$, $\mathrm{S}\mathrm{p}(2m,\mathbb{C})\subset \mathrm{S}\mathrm{p}(2n,\mathbb{C})$, $\mathrm{S}\mathrm{O}(q,\mathbb{C})\subset \mathrm{S}\mathrm{O}(p,\mathbb{C})$ for all $m\le n$, and all $q\le p$, 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.