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

M. Nazarov; Vitaly Tarasov

doi:10.1215/S0012-9074-02-11225-3

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

M. Nazarov, Vitaly Tarasov

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We study the tensor product W of any number of irreducible finite-dimensional modules V-1, ..., V-k over the Yangian Y(gl(N)) of the general linear Lie algebra gl(N). For any indices i, j = 1, ..., k, there is a canonical nonzero intertwining operator J(ij) : V-i circle times V-j --> V-j circle times V-i. It has been conjectured that the tensor product W is irreducible if and only if all operators Jij with i < j are invertible. We prove this conjecture for a wide class of irreducible Y(gl(N))-modules V-1, ..., V-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.

Original language	English
Pages (from-to)	343-378
Number of pages	36
Journal	Duke Mathematical Journal
Volume	112
Issue number	2
DOIs	https://doi.org/10.1215/S0012-9074-02-11225-3
Publication status	Published - 1 Apr 2002

Keywords

XXZ MODEL
R-MATRIX
REPRESENTATIONS
ALGEBRAS
BASES

Access to Document

10.1215/S0012-9074-02-11225-3

http://dx.doi.org/10.1215/S0012-9074-02-11225-3

Cite this

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title = "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-1, ..., V-k over the Yangian Y(gl(N)) of the general linear Lie algebra gl(N). For any indices i, j = 1, ..., k, there is a canonical nonzero intertwining operator J(ij) : V-i circle times V-j --> V-j circle times V-i. It has been conjectured that the tensor product W is irreducible if and only if all operators Jij with i < j are invertible. We prove this conjecture for a wide class of irreducible Y(gl(N))-modules V-1, ..., V-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.",

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T1 - On irreducibility of tensor products of Yangian modules associated with skew Young diagrams

AU - Nazarov, M.

AU - Tarasov, Vitaly

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N2 - We study the tensor product W of any number of irreducible finite-dimensional modules V-1, ..., V-k over the Yangian Y(gl(N)) of the general linear Lie algebra gl(N). For any indices i, j = 1, ..., k, there is a canonical nonzero intertwining operator J(ij) : V-i circle times V-j --> V-j circle times V-i. It has been conjectured that the tensor product W is irreducible if and only if all operators Jij with i < j are invertible. We prove this conjecture for a wide class of irreducible Y(gl(N))-modules V-1, ..., V-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.

AB - We study the tensor product W of any number of irreducible finite-dimensional modules V-1, ..., V-k over the Yangian Y(gl(N)) of the general linear Lie algebra gl(N). For any indices i, j = 1, ..., k, there is a canonical nonzero intertwining operator J(ij) : V-i circle times V-j --> V-j circle times V-i. It has been conjectured that the tensor product W is irreducible if and only if all operators Jij with i < j are invertible. We prove this conjecture for a wide class of irreducible Y(gl(N))-modules V-1, ..., V-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.

KW - XXZ MODEL

KW - R-MATRIX

KW - REPRESENTATIONS

KW - ALGEBRAS

KW - BASES

U2 - 10.1215/S0012-9074-02-11225-3

DO - 10.1215/S0012-9074-02-11225-3

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