Young diagrammatic methods for the restriction of representations of complex classical Lie groups to reductive subgroups of maximal rank
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 and other such tableaux have been described in a recent review [\textit{S. Sundaram} in Invariant theory and tableaux, Proc. Workshop, Minneapolis/MN (USA) 1988, IMA Vol. Math. Appl. 19, 191--225 (1989; Zbl 0707.22004)].
scientific article; zbMATH DE number 4144294
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