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.


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


Zbl 0722.00027