## On irreducibility of tensor products of Yangian modules associated with skew Young diagrams.(English)Zbl 1027.17013

In this paper the authors continue their study [begun in J. Reine Angew. Math. 496, 181-212 (1998; Zbl 0885.17009)] of the tensor product $$W$$ of any number of irreducible finite-dimensional modules $$V_1,\dots,V_k$$ over the Yangian $$Y(\mathfrak{gl}_N)$$ of the general linear Lie algebra $$\mathfrak{gl}_N$$. For any indices $$i,j=1,\dots,k$$, there is a canonical nonzero intertwining operator $$J_{ij}\colon V_i\otimes V_j \to V_j\otimes V_i$$. It has been conjectured [see, for instance, V. Chari and A. Pressley, J. Reine Angew. Math. 417, 87-128 (1991; Zbl 0726.17014)] that the tensor product $$W$$ is irreducible if and only if all operators $$J_{ij}$$ with $$i<j$$ are invertible. The authors prove this conjecture for a wide class of irreducible $$Y(\mathfrak{gl}_N)$$-modules $$V_1,\dots,V_k$$. Each of these modules is determined by a skew Young diagram and a complex parameter. This result implies, in particular, that $$W$$ is irreducible if and only if for all $$i<j$$ the pairwise tensor products $$V_i\otimes V_j$$ are irreducible. The authors 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.

### MSC:

 17B37 Quantum groups (quantized enveloping algebras) and related deformations

### Citations:

Zbl 0885.17009; Zbl 0726.17014
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