Abstract
A new combinatorial expression is given for the dimension of the space of invariants in the tensor product of three irreducible finite dimensional sl(r + 1)-modules (we call this dimension the triple multiplicity). This expression exhibits a lot of symmetries that are not clear from the classical expression given by the Littlewood–Richardson rule. In our approach the triple multiplicity is given as the number of integral points of the section of a certain “universal” polyhedral convex cone by a plane determined by three highest weights. This allows us to study triple multiplicities using ideas from linear programming. As an application of this method, we prove a conjecture of B. Kostant that describes all irreducible constituents of the exterior algebra of the adjoint sl(r + 1)-module.
Article PDF
Similar content being viewed by others
References
A.D. Berenstein and A.V. Zelevinsky, “Involutions on Gelfand-Tsetlin patterns and multiplicities in skew gℓn-modules,” Doklady AN SSSR, vol. 300, no. 6, pp. 1291–1294, 1988 (in Russian).
A.D. Berenstein and A.V. Zelevinsky, “Tensor product multiplicities and convex polytopes in partition space,” Journal of Geometry and Physics, vol. 5, no. 3, pp. 453–472, 1988.
N. Bourbaki, Groupes et algèbres de Lie, Ch. IV, V, VI. Hermann: Paris, 1968.
C. Carré, “Le décodage de la régle de Littlewood-Richardson dans les triangles de Berenstein-Zelevinsky,” preprint, April 1991.
C. Davis, “Theory of positive linear dependence,” American Journal of Mathematics, vol. 76, pp. 733–746, 1954.
D. Gale, The theory of linear economic models, McGraw-Hill, New York, 1960.
I.M. Gelfand and A.V. Zelevinsky, “Polytopes in the pattern space and canonical basis in irreducible representations of gℓ3,” Functional Analysis and Applications, vol. 19, no. 2, pp. 72–75, 1985 (in Russian).
I.M. Gelfand and A.V. Zelevinsky, “Multiplicities and regular bases for gℓn,” in Group theoretical methods in physics, Proc. of the third seminar, Yurmala, May 22–24, 1985. Nauka, Moscow, vol. 2, pp. 22–31, 1986 (in Russian).
I.M. Gelfand, A.V. Zelevinsky, and M.M. Kapranov, “Newton polytopes of the classical resultant and discriminant,” Advances in Mathematics, vol. 84, no. 2, pp. 237–254, 1990.
I. Macdonald, Symmetric functions and Hall polynomials, Clarendon Press, Oxford, 1979.
E. Verlinde, “Fusion rule and modular transformation in 2d conformal field theory,” Nuclear Physics B, vol. 300, pp. 360–376, 1988.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Berenstein, A., Zelevinsky, A. Triple Multiplicities for sl(r + 1) and the Spectrum of the Exterior Algebra of the Adjoint Representation. Journal of Algebraic Combinatorics 1, 7–22 (1992). https://doi.org/10.1023/A:1022429213282
Issue Date:
DOI: https://doi.org/10.1023/A:1022429213282