Symplectic tableaux and quantum symmetric pairs

Abstract

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 $({\mathfrak{gl}}_{2n}(\mathbb{C}),{\mathfrak{sp}}_{2n}(\mathbb{C}))$.