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

By M. Nazarov, V. Tarasov. 2000

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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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3f0b0a9e47bee55a824500cce6154bb78b9e67b6

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In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Quantum Integrability and Supersymmetric Vacua (2009, methodology): “Similar phenomena occur in other models, at Q → 0 in the XXX model [31], in the XXZ model with e being a root of unity, in the six-vertex model [32], etc.”