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

Nazarov, Maxim; Tarasov, Vitaly

Duke Math. J. 112, No. 2, 343-378 (2002).

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.

Reviewer: Mirko Primc (Zagreb)

MSC:

17B37

Quantum groups (quantized enveloping algebras) and related deformations

Keywords:

tensor product; Yangian; general linear Lie algebra; interwining operator; Durfee rank; skew Young diagram

Citations:

Zbl 0885.17009; Zbl 0726.17014