Symplectic tableaux and quantum symmetric pairs

We provide a new branching rule from the general linear group \mathrm{GL}_{2n}(\mathbb{C}) to the symplectic group \mathrm{Sp}_{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{II}_{2n-1} , which is a q -analogue of the classical symmetric pair (\mathfrak{gl}_{2n}(\mathbb{C}), \mathfrak{sp}_{2n}(\mathbb{C})) .