Symplectic tableaux and quantum symmetric pairs

TL;DR

This paper introduces a new combinatorial branching rule for the symplectic group derived from quantum symmetric pair theory, providing an algorithmic bijection involving symplectic tableaux.

Contribution

It presents a novel algorithmic bijection linking semistandard and symplectic tableaux, based on quantum symmetric pair representation theory.

Findings

Provides a simple algorithm for the branching rule

Establishes a bijection between tableaux sets

Connects quantum symmetric pairs with classical representation theory

Abstract

We provide a new branching rule from the general linear group $GL_{2n}(\mathbb{C})$ to the symplectic group $Sp_{2n}(\mathbb{C})$ 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 $(\mathfrak{gl}_{2n}(\mathbb{C}), \mathfrak{sp}_{2n}(\mathbb{C}))$.