Young diagrammatic methods for the restriction of representations of complex classical Lie groups to reductive subgroups of maximal rank
type: Item
instance of: scholarly article
title: Young diagrammatic methods for the restriction of representations of complex classical Lie groups to reductive subgroups of maximal rank
zbMATH Open document ID: 0698.22013
DOI: 10.1016/0001-8708(90)90059-V
author: Kazuhiko Koike
author: Itaru Terada
published in: Advances in Mathematics
publication date: 1990-01-01T00:00:00Z
review text: The authors provide restriction formulae \(\lambda_ G\downarrow^ G_ H\) for irreducible representations \(\lambda_ G\) of the complex classical groups \(G\) with respect to a number of reductive subgroups \(H\) of maximal rank, including the Levi subgroups \(L(P)\) of parabolic subgroups \(P\). The results are derived using methods based on universal characters [the authors, J. Algebra 107, 466--511 (1987; Zbl 0622.20033); \textit{R. C. King} in Invariant theory and tableaux, Proc. Workshop, Minneapolis/MN (USA) 1988, IMA Vol. Math. Appl. 19, 226--261 (1989; Zbl 0713.17006)]. They are expressed in terms of the Littlewood-Richardson coefficients governing products and quotients of Schur functions [\textit{D. E. Littlewood}, The theory of group characters and matrix representations of groups. London: Oxford Univ. Press (1940; Zbl 0025.00901, JFM 66.0093.02); \textit{I. G. Macdonald}, Symmetric functions and Hall polynomials Oxf. Univ. Press (1979; Zbl 0487.20007)] and certain specialization homomorphisms [the authors, op. cit.]. Formulas are given for the restrictions: \(\text{SO}(2n+1)\to \text{GL}(k)\times \text{SO}(2n+1-2k)\); \(\text{Sp}(2n)\to \text{GL}(k)\times \text{Sp}(2n-2k)\); \(\text{SO}(2n)\to \text{GL}(k)\times \text{SO}(2n-2k)\); \(\text{SO}(2n+1)\to \text{SO}(2k)\times \text{SO}(2n+1-2k)\); \(\text{Sp}(2n)\to \text{Sp}(2k)\times \text{Sp}(2n-2k)\); \(\text{SO}(2n)\to \text{SO}(2k)\times \text{SO}(2n-2k)\); \(\text{SO}(2n+1)\to \text{GL}(n)\); \(\text{Sp}(2n)\to \text{GL}(n)\); \(\text{SO}(2n)\to \text{GL}(n)\). The first three restriction rules are new, but can be obtained from results given elsewhere [\textit{R. C. King}, J. Phys. A 8, 429--449 (1975; Zbl 0301.20029)] which explicitly include the remaining six restriction rules given here. Special cases of the restriction rules are then exploited to construct sets of standard Young tableaux appropriate to irreducible representations of \(\text{SO}(2n+1)\), \(\text{Sp}(2n)\) and \(\text{SO}(2n)\). These…
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 20G20
zbMATH DE Number: 4144294
zbMATH Keywords: restriction formulae
zbMATH Keywords: irreducible representations
zbMATH Keywords: complex classical groups
zbMATH Keywords: reductive subgroups
zbMATH Keywords: maximal rank
zbMATH Keywords: Levi subgroups
zbMATH Keywords: parabolic subgroups
zbMATH Keywords: universal characters
zbMATH Keywords: Littlewood-Richardson coefficients
zbMATH Keywords: Schur functions
zbMATH Keywords: restriction rules
zbMATH Keywords: standard Young tableaux
Wikidata QID: Q115368881
reviewed by: Ronald C. King
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4144294