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

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.