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Triple multiplicities for $$s\ell (r+1)$$ and the spectrum of the exterior algebra of the adjoint representation. (English) Zbl 0799.17005
The triple multiplicity is the dimension of the space of invariant vectors in a tensor product of three irreducible finite dimensional representations of the Lie algebra $$sl (n)$$. The authors give a geometric interpretation to these multiplicities. The triple multiplicity is equal to the number of integral points of the section of a certain “universal” polyhedral convex cone by a plane determined by three highest weights. This interpretation allows to study triple multiplicities using ideas from the theory of linear programming. By making use of this method, the authors prove the conjecture of B. Kostant which describes all irreducible constituents of the exterior algebra of the adjoint representation of the Lie algebra $$sl (n)$$. The approach to triple multiplicities described in this paper exhibits many of their symmetries which can not be obtained from the expression given by the Littlewood-Richardson rule.
Reviewer: A.Klimyk (Kiev)

##### MSC:
 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 22E46 Semisimple Lie groups and their representations 05E15 Combinatorial aspects of groups and algebras (MSC2010)
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##### References:
 [1] Berenstein, A. D.; Zelevinsky, A. V., Involutions on Gelfand-Tsetlin patterns and multiplicities in skew $$g$$ ℓ_{n}-modules, Doklady AN SSSR, 300, 1291-1294, (1988) [2] Berenstein, A. D.; Zelevinsky, A. V., Tensor product multiplicities and convex polytopes in partition space, Journal of Geometry and Physics, 5, 453-472, (1988) · Zbl 0712.17006 [3] N. Bourbaki, Groupes et algèbres de Lie, Ch. IV, V, VI. Hermann: Paris, 1968. [4] C. Carré, “Le décodage de la régle de Littlewood-Richardson dans les triangles de Berenstein-Zelevinsky,” preprint, April 1991. [5] Davis, C., Theory of positive linear dependence, American Journal of Mathematics, 76, 733-746, (1954) · Zbl 0058.25201 [6] D. Gale, The theory of linear economic models, McGraw-Hill, New York, 1960. [7] Gelfand, I. M.; Zelevinsky, A. V., Polytopes in the pattern space and canonical basis in irreducible representations of $$g$$ ℓ_{3}, Functional Analysis and Applications, 19, 72-75, (1985) [8] Gelfand, I. M.; Zelevinsky, A. V., Multiplicities and regular bases for $$g$$ ℓ_{n}, 22-31, (1986), Moscow [9] Gelfand, I. M.; Zelevinsky, A. V.; Kapranov, M. M., Newton polytopes of the classical resultant and discriminant, Advances in Mathematics, 84, 237-254, (1990) · Zbl 0721.33002 [10] I. Macdonald, Symmetric functions and Hall polynomials, Clarendon Press, Oxford, 1979. · Zbl 0487.20007 [11] Verlinde, E., Fusion rule and modular transformation in 2d conformal field theory, Nuclear Physics B, 300, 360-376, (1988) · Zbl 1180.81120
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