Kashiwara--Nakashima tableaux, Gelfand--Tsetlin patterns, and quantum symmetric pairs

Kashiwara–Nakashima tableaux and Gelfand–Tsetlin patterns of orthogonal type are famous as combinatorial models of the finite-dimensional irreducible representations of the special orthogonal Lie algebras ${\mathfrak{s}\mathfrak{o}}_N$. The former is constructed based on the representation theory of quantum groups, especially the theory of crystals, while the latter based on the branching rule for $({\mathfrak{s}\mathfrak{o}}_N,{\mathfrak{s}\mathfrak{o}}_{N-1})$. In the present paper, we construct a natural bijection between them by means of the representation theory of quantum symmetric pairs corresponding to $({\mathfrak{s}\mathfrak{o}}_N,{\mathfrak{s}\mathfrak{o}}_{N-1})$.