Representations of twisted Yangians associated with skew Young diagrams

Let GM be either the orthogonal group OM , or the symplectic group SpM over the complex field; in the latter case the non-negative integer M has to be even. Classically, the irreducible finite-dimensional representations of the group GM are labeled by partitions μ = (μ1, μ2 , . . .) such that μ ′ 1+μ ′ 2 6 M in the case GM = OM , or 2μ ′ 1 6 M in the case GM = SpM . Here μ ′ = (μ ′ 1, μ ′ 2 , . . .) is the partition conjugate to μ. Let Wμ be the irreducible finite-dimensional representation of the group GM corresponding to μ. Regard the direct product GN ×GM as a subgroup of GN+M . Take any irreducible finite-dimensional representation Wλ of the group GN+M . Consider the vector space Wλ(μ) = HomGM (Wμ ,Wλ ), it comes with a natural action of the group GN . Put n = λ1 − μ1 + λ2 − μ2 + . . . . In this article, for any standard Young tableau Ω of skew shape λ/μ we give a realization of Wλ(μ) as a subspace in the n -fold tensor product (C ), compatible with the action of the group GN . This subspace is determined as the image of a certain linear operator FΩ (M) on (C N ), given by an explicit formula. When M = 0 and Wλ(μ) = Wλ is an irreducible representation of the group GN , we recover the classical realization of Wλ as a subspace in the space of all traceless tensors in (C ). Then the operator FΩ (0) may be regarded as the analogue for GN of the Young symmetrizer, corresponding to the standard tableau Ω of shape λ . This symmetrizer is a certain linear operator on (C) with the image equivalent to the irreducible polynomial representation of the complex general linear group GLN , corresponding to the partition λ . Even in the case M = 0, our formula for the operator FΩ (M) is new. Our results are applications of the representation theory of the twisted Yangian, corresponding to the subgroup GN of GLN . This twisted Yangian is a certain one-sided coideal subalgebra of the Yangian corresponding to GLN . In particular, FΩ (M) is an intertwining operator between certain representations of the twisted Yangian in (C ). This work may be regarded as a continuation of the classical works of Gelfand and Zetlin on the constructions of irreducible representations of the groups GLN and ON . Mathematics Subject Classification (2000). 17B35, 17B37, 20C30, 22E46, 81R50.