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Tensor product multiplicities and convex polytopes in partition space. (English) Zbl 0712.17006
An important theoretical problem, and one which has physical applications, is the determination of weight multiplicities in the reduction of the tensor product of two irreducible finite-dimensional representations of a semisimple complex Lie algebra. The approach adopted in this article is based on the hierarchical relationship between the Kostant partition function, the weight multiplicity function for weights of an irreducible representation and the tensor product multiplicity function. Recalling that the Kostant partition function for a given weight is the number of partitions of it into a sum of positive roots, and that this can be interpreted as the number of integral points of a convex polytope in partition space, the authors conjecture that an analogous result should be true for the multiplicity functions. A very precise conjecture is made for classical Lie algebras and proved for algebras of small rank. Finally, reduction multiplicities are studied in terms of generalized Gelfand-Tsetlin patterns.

MSC:
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B20 Simple, semisimple, reductive (super)algebras
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