On the decomposition of tensor products of the representations of the classical groups: By means of the universal characters
type: Item
instance of: scholarly article
title: On the decomposition of tensor products of the representations of the classical groups: By means of the universal characters
zbMATH Open document ID: 0681.20030
DOI: 10.1016/0001-8708(89)90004-2
author: Kazuhiko Koike
published in: Advances in Mathematics
publication date: 1989-01-01T00:00:00Z
review text: Let \(\Lambda_ x\) be the projective limit of graded algebra \({\mathbb{Z}}[x_ 1,x_ 2,...,x_ n]^{S_ n}\) of symmetric polynomials in n variables. The author defines the universal characters for any rational irreducible representation of the general linear group GL(n,\({\mathbb{C}})\) (resp. of the symplectic group Sp(2n,\({\mathbb{C}})\) or of the special orthogonal group SO(n,\({\mathbb{C}}))\). These universal characters are certain elements of \(\Lambda_ x\otimes \Lambda_ y\) which can be expressed as the determinants of some elementary symmetric functions or of some symmetric power sums. These universal characters can be specialized into the irreducible characters of GL(n,\({\mathbb{C}})\) (resp. of Sp(2n,\({\mathbb{C}})\) or of SO(n,\({\mathbb{C}}))\) under certain specialization homomorphism. In the paper under review, the author gives the decomposition formulas of tensor products between the universal characters in terms of Littlewood- Richardson coefficients. From this, he gives the decomposition of the tensor product \(V_{\lambda}\otimes V^*_{\mu}\), where \(V_{\eta}\) is the irreducible representation space of GL(n,\({\mathbb{C}})\) with the highest weight \(\eta\), \(V^*_{\eta}\) is the dual space of \(V_{\eta}\). On the other hand, the author proves the decomposition formulas of the tensor product of two irreducible representations of Sp(2n,\({\mathbb{C}})\) and of SO(n,\({\mathbb{C}})\) by making use of the corresponding universal characters. Comparing with the formula given by the author in his previous paper [the author and \textit{I. Terada}, J. Algebra 107, 466-511 (1987; Zbl 0622.20033)], the present formulas have the advantage that all of their coefficients are non-negative and hence they can be calculated easier.
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G20
zbMATH DE Number: 4114886
zbMATH Keywords: projective limit of graded algebra
zbMATH Keywords: symmetric polynomials
zbMATH Keywords: universal characters
zbMATH Keywords: rational irreducible representation
zbMATH Keywords: general linear group
zbMATH Keywords: symplectic group
zbMATH Keywords: special orthogonal group
zbMATH Keywords: irreducible characters
zbMATH Keywords: decomposition formulas of tensor products
MaRDI profile type: Publication
full work available at URL: 0001-8708(89)90004-2
OpenAlex ID: W2092309185
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description: scientific article; zbMATH DE number 4114886