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The Littlewood-Richardson rule – the cornerstone for computing group properties. (English) Zbl 0747.22011

Topics in algebra. Pt. 2: Commutative rings and algebraic groups, Pap. 31st Semester Class. Algebraic Struct., Warsaw/Poland 1988, Banach Cent. Publ. 26, Part 2, 475-482 (1990).

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[For the entire collection see Zbl 0716.00007.]
It is pointed out that labelling irreducible representations of both classical and exceptional Lie groups by means of partitions and using \(S\)-functions allows applications of the Littlewood-Richardson rule to be extended far beyond its original symmetric group context [D. E. Littlewood and A. R. Richardson, Philos. Trans. R. Soc. Lond., Ser. A 233, 99-141 (1934; Zbl 0009.20203)]. Its implementation in SCHUR [SCHUR Software Associates, Christchurch, New Zealand] leads to the efficient interactive evaluation of a large variety of such things as tensor products and branching rules for Lie groups. Applications to atomic, nuclear and particle physics are mentioned, and a few illustrative examples involving infinite series of \(S\)-functions, skew \(S\)-functions, plethysms and user defined functions are given.

MSC:

22E70 Applications of Lie groups to the sciences; explicit representations
05E05 Symmetric functions and generalizations
20C35 Applications of group representations to physics and other areas of science
05E10 Combinatorial aspects of representation theory
20C40 Computational methods (representations of groups) (MSC2010)
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