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A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras. (English) Zbl 0805.17019
The Littlewood-Richardson rule to decompose the tensor product of two irreducible \(\text{sl}(n,\mathbb C)\)-modules into irreducible components is well-known. In this paper, for a symmetrizable Kac-Moody Lie algebra \({\mathfrak g}\), the author has obtained a Littlewood-Richardson type rule to decompose the tensor product of two integrable highest weight \({\mathfrak g}\)-modules into irreducible components using certain piecewise linear paths which he names Lakshmibai-Seshadri paths. As applications of this decomposition rule, (1) he has given another proof of the well-known PRV conjecture and, (2) he has obtained a branching rule for the restriction of the integrable highest weight \({\mathfrak g}\)-module to a Levi subalgebra of \({\mathfrak g}\) in terms of the Lakshmibai-Seshadri paths.

MSC:
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B20 Simple, semisimple, reductive (super)algebras
17B37 Quantum groups (quantized enveloping algebras) and related deformations
20G05 Representation theory for linear algebraic groups
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