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Representations of \(U_ q({\mathfrak gl}(n,\mathbb{C} ))\) at \(q=0\) and the Robinson-Schensted correspondence. (English) Zbl 0743.17018

Physics and mathematics of strings, Mem. Vol. Vadim Knizhnik, 185-211 (1990).
[For the entire collection see Zbl 0722.00027.]
From the author’s abstract: “Let \(V\) be the \(n\) dimensional irreducible \(U_ q({\mathfrak gl}(n,\mathbb{C}))\)-module and let \(v_ \mu (\mu=1,\ldots,n)\) be it natural basis. There is an irreducible decomposition \(V^{\otimes N}=\oplus_ T V(T)\) where \(T\) is a standard tableau with \(N\) nodes. It is shown that in the \(q^{\pm 1}\to 0\) limit \(V(T)\) is spanned by decomposable vectors \(v_{\mu_ 1}\otimes\cdots\otimes v_{\mu_ N}\), and that the correspondence \(T\leftrightarrow v_{\mu_ 1}\otimes\cdots\otimes v_{\mu_ N}\) is given by the Robinson-Schensted correspondence.”
Although \(U_ q({\mathfrak gl}(n,\mathbb{C}))\) is not defined for \(q=0\), the authors make precise what they mean by the limits \(q\to 0\) and \(q^{- 1}\to 0\), which involves the vanishing of certain reduced Wigner coefficients. The authors say that their motivation comes from the computation of one-point functions in solvable lattice models. In this paper, they calculate the spectral decomposition of the \(R\)-matrix associated with the \(N\)-th symmetric tensor representation \(V_{[N]}\), and determine its \(H\)-function by using the Robinson-Schensted description. It results that the one-point functions for the \(N\)-th symmetric tensor model are expressible in terms of the level \(N\) string functions.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
05E10 Combinatorial aspects of representation theory

Biographic References:

Knizhnik, Vadim

Citations:

Zbl 0722.00027
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