Berele row-insertion and quantum symmetric pairs

The Berele row-insertion is a symplectic analogue of the Schensted row-insertion. In the present paper, we provide it with a representation theoretical interpretation via the quantum symmetric pairs of type $A\mathrm{I}\mathrm{I}$. As applications, we lift Berele’s Robinson–Schensted correspondence and Kobayashi–Matsumura’s Robinson–Schensted–Knuth (RSK for short) correspondence to isomorphisms of representations over a quantum symmetric pair coideal subalgebra, and establish the dual RSK correspondence of type $A\mathrm{I}\mathrm{I}$.