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Rational representations of Yangians associated with skew Young diagrams

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Abstract.

Consider the general linear group GL M over the complex field. The irreducible rational representations of the group GL M can be labeled by the pairs of partitions and such that the total number of non-zero parts of μ and does not exceed M. Let EQ4 be the irreducible representation corresponding to such a pair. Regard the direct product as a subgroup of GL N+M . Take any irreducible rational representation of GL N+M . The vector space comes with a natural action of the group GL N . 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 GL N , and of ñ copies of the contragredient representation ()*. This subspace is determined as the image of a certain linear operator on W nñn . We introduce this operator by an explicit multiplicative formula. When M=0 and is an irreducible representation of GL N , 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 Ω of shape λ . 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.

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Authors and Affiliations

  1. Department of Mathematics, University of York, York, YO10 5DD, England

    Maxim Nazarov

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  1. Maxim Nazarov
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Correspondence to Maxim Nazarov.

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Mathematics Subject Classification (2000): 17B37, 20C30, 22E46, 81R50

in final form: 10 July 2003

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Nazarov, M. Rational representations of Yangians associated with skew Young diagrams. Math. Z. 247, 21–63 (2004). https://doi.org/10.1007/s00209-003-0619-7

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  • DOI: https://doi.org/10.1007/s00209-003-0619-7

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