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

Nazarov, M. and Tarasov, V. (2002) On irreducibility of tensor products of Yangian modules associated with skew Young diagrams. Duke Mathematical Journal, 112 (2). pp. 343-378. ISSN 0012-7094

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## Abstract

We study the tensor product $W$ of any number of irreducible finite-dimensional modules $V\sb 1,\ldots V\sb k$ over the Yangian ${\rm Y}(\mathfrak {gl}\sb N)$ of the general linear Lie algebra $\mathfrak {gl}\sb N$. For any indices $i,j=1,\ldots k$, there is a canonical nonzero intertwining operator $J\sb {ij} : V\sb i\otimes V\sb j\to V\sb j\otimes V\sb i$. It has been conjectured that the tensor product $W$ is irreducible if and only if all operators $J\sb {ij}$ with $i<j$ are invertible. We prove this conjecture for a wide class of irreducible ${\rm Y}(\mathfrak {gl}\sb N)$-modules $V\sb 1,\ldots V\sb k$. Each of these modules is determined by a skew Young diagram and a complex parameter. 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.

Item Type: Article The University of York The University of York > Mathematics (York) York RAE Import 15 Jun 2009 14:36 15 Jun 2009 14:36 http://dx.doi.org/10.1215/S0012-9074-02-11225-3 Published Duke University Press 10.1215/S0012-9074-02-11225-3 http://eprints.whiterose.ac.uk/id/eprint/5983