Rational representations of Yangians associated with skew Young diagrams

Consider the general linear group GLM over the complex field. The irreducible rational representations of the group GLM can be labeled by the pairs of partitions and such that the total number of non-zero parts of and does not exceed M. Let EQ4 be the irreducible representation corresponding to such a pair. Regard the direct product as a subgroup of GLN+M . Take any irreducible rational representation of GLN+M. The vector space comes with a natural action of the group GLN. Put n=. For any pair of standard Young tableaux of skew shapes respectively, we give a realization of as a subspace in the tensor product of n copies of defining representation of GLN, and of ñ copies of the contragredient representation ()*. This subspace is determined as the image of a certain linear operator on Wnñn. We introduce this operator by an explicit multiplicative formula. When M=0 and is an irreducible representation of GLN, we recover the known realization of as a certain subspace in the space of all traceless tensors in . Then the operator may be regarded as the rational analogue of the Young symmetrizer, corresponding to the tableau of shape . Even when M=0, our formula for is new. Our results are applications of the representation theory of the Yangian of the Lie algebra . In particular, is an intertwining operator between certain representations of the algebra on . We also introduce the notion of a rational representation of the Yangian . As a representation of , the image of is rational and irreducible.