This article draws on the ideas beginning with the Schur-Weyl duality, passing through Gelfand-Zetlin bases and continuing through Yangians, with the aim of realizing explicitly irreducible representations of the orthogonal or symplectic classical group \(G_N\). This work provides new explicit realization of the representation of \(G_N\) on the vector space. This vector space describes the multiplicities in the restriction of an irreducible representation of the group \(G_{N+M}\) to the subgroup \(G_M\). The results follow for the branching rules for restricting irreducible representations from \(G_{N+M}\) to the subgroup \(G_{N}\times G_M\).
An overview of this article is as follows. Section 1 gives an exposition of the principal results. Section 2 recalls the classical realization of any irreducible polynomial representation of the general linear group \(\text{GL}_N\) and the approach to Young symmetrizers. Following this approach, in Section 3, the author constructs analogues of the Young symmetrizers for the group \(G_N\). This construction provides a realization of any irreducible polynomial representation of the group \(G_N\). It is motivated by the representation theory of Yangians and of their twisted analogues. The main results concerning branching rules for the groups \(\text{GL}_N\) and \(G_N\) are stated as Theorems 1.6 and 1.8 respectively. Theorem 1.6, which says that the \(Y(\text{gl}_N)\)-modules \(V_{\lambda}(\mu)\) and \(V_{\Omega}\) are equivalent, belongs to I. V. Cherednik [Duke Math. J. 54, 563–577 (1987; Zbl 0645.17006)], but its proof given in this paper is new. Theorem 1.8 is a contribution of the author.