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

2000/12/07 by Maxim Nazarov, Vitaly Tarasov · 1 citation
Mathematics · #math.QA #math.CO #math.RT

paper · pdf

published as Duke Math. J. 112 (2002), 343-378 · 29 pages, AmS-TeX, with additions to Section 4

arxiv created 2000/12/07 · arxiv updated 2009/11/30

Abstract

We study the tensor product W of any number of "elementary" irreducible modules V₁,...,V_k over the Yangian of the general linear Lie algebra. Each of these modules is determined by a skew Young diagram and a complex parameter. For any indices i,j=1,...,k there is a canonical non-zero intertwining operator A_ij between the tensor products V_i⊗ V_j and V_j⊗ V_i. This operator is defined up to a scalar multipler. We show that the tensor product W is irreducible, if and only if all operators A_ij with i<j are invertible. This implies that the Yangian module W is irreducible, if and only if all pairwise tensor products V_i⊗ V_j with i<j are irreducible. 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.

Cited by

• A generalization of Carter-Payne homomorphisms

Related

• Rational representations of Yangians associated with skew Young diagrams
• Irreducibility criterion for tensor products of Yangian evaluation modules
• Representations of twisted Yangians associated with skew Young diagrams
• Finite-dimensional Representations of Yangians
• Irreducibility of the tensor product of Yangian \( Y(\mathfrakgl_m|n) \) evaluation modules