Symplectic tableaux and quantum symmetric pairs

We provide a new branching rule from the general linear group $G{L}_{2n}(\mathbb{C})$ to the symplectic group $S{p}_{2n}(\mathbb{C})$ by establishing a simple algorithm which gives rise to a bijection from the set of semistandard tableaux of a fixed shape to a disjoint union of several copies of sets of symplectic tableaux of various shapes. The algorithm arises from representation theory of a quantum symmetric pair of type $A{\mathrm{I}\mathrm{I}}_{2n-1}$, which is a $q$-analogue of the classical symmetric pair $({\mathfrak{g}\mathfrak{l}}_{2n}(\mathbb{C}),{\mathfrak{s}\mathfrak{p}}_{2n}(\mathbb{C}))$.