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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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##### References:
 [1] Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. Berlin Heidelberg New York: Springer 1968 · Zbl 0254.17004 [2] Kac, V.: Infinite-dimensional Lie-algebras. Cambridge: Cambridge University Press 1985 · Zbl 0574.17010 [3] Kashiwara, M.: Crystalizing theq-analogue of Universal Enveloping algebras. Commun. Math. Phys.133, 249-260 (1990) · Zbl 0724.17009 · doi:10.1007/BF02097367 [4] Kashiwara, M.: Crystalizing theq-analogue of Universal Enveloping algebras, Preprint, Res. Inst. Math. Sci.133 (1990) · Zbl 0724.17009 [5] Kashiwara, M.: Crystal base and Littelmann’s refined Demazure character formula. Preprint. Res. Inst. Math. Sci.133 (1992) · Zbl 0794.17008 [6] Klimyk, A.U.: Decomposition of a tensor product of irreducible representations of a semi-simple Lie algebra into a direct sum of irreducible representations. Transl., II. Ser., Am. Math. Soc.76, 63-73 (1968) · Zbl 0228.17004 [7] Kumar, S.: Proof of the Parthasarathy-Ranga-Rao-Varadarajan Conjecture. Invent. Math.93, 117-130 (1988) · Zbl 0668.17008 · doi:10.1007/BF01393689 [8] Kumar, S.: Demazure character formula in arbitrary Kac-Moody setting. Invent. Math.89, 395-423 (1987) · Zbl 0635.14023 · doi:10.1007/BF01389086 [9] Lakshmibai, V., Seshadri, C.S.: Standard monomial theory. In: Proceedings of the Hyderabad Conference on Algebraic Groups, pp. 279-323. Madras: Manoj Prakashan 1991 · Zbl 0785.14028 [10] Lakshmibai, V., Seshadri, C.S.: Geometry ofG/P V. J. Algebra100, 462-557 (1986). · Zbl 0618.14026 · doi:10.1016/0021-8693(86)90089-X [11] Littelmann, P.: A generalization of the Littlewood-Richardson rule. J. Algebra130, 328-368 (1990) · Zbl 0704.20033 · doi:10.1016/0021-8693(90)90086-4 [12] Littelmann, P.: Young tableaux and crystal bases. (Preprint); J. Algebra (to appear, 1992) [13] Mathieu, O.: Construction d’un groupe de Kac-Moody et applications. Compos. Math.69, 37-60 (1989) · Zbl 0678.17012 [14] Mathieu, O.: Formules de caractères pour les algèbres de Kac-Moody générales. Astérisque159-160 (1988)
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