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

Abstract

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 $\mathfrak{so}_N$. 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 $(\mathfrak{so}_N,\mathfrak{so}_{N-1})$. In the present paper, we construct a natural bijection between them by means of the representation theory of quantum symmetric pairs corresponding to $(\mathfrak{so}_N,\mathfrak{so}_{N-1})$.