On irreducibility of tensor products of Yangian modules associated with skew Young diagrams

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We study the tensor product W of any number of "elementary" irreducible modules V₁,...,Vₖ over the Yangian of the general linear Lie algebra. Each of these modules is determined by a skew Young diagram and a complex parameter. For any indices i,j=1,...,k there is a canonical non-zero intertwining operator Aij between the tensor products Vᵢ⊗ Vⱼ and Vⱼ⊗ Vᵢ. This operator is defined up to a scalar multipler. We show that the tensor product W is irreducible, if and only if all operators Aij with i<j are invertible. This implies that the Yangian module W is irreducible, if and only if all pairwise tensor products Vᵢ⊗ Vⱼ with i<j are irreducible. We also introduce the notion of a Durfee rank of a skew Young diagram. For an ordinary Young diagram, this is the length of its main diagonal.