Journal of Algebraic Combinatorics

, Volume 1, Issue 1, pp 7–22

Triple Multiplicities for sl(r + 1) and the Spectrum of the Exterior Algebra of the Adjoint Representation

  • A.D. Berenstein
  • A.V. Zelevinsky
Article

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.

tensor product multiplicities systems of linear inequalities 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A.D. Berenstein and A.V. Zelevinsky, “Involutions on Gelfand-Tsetlin patterns and multiplicities in skew gn-modules,” Doklady AN SSSR, vol. 300, no. 6, pp. 1291–1294, 1988 (in Russian).Google Scholar
  2. 2.
    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.Google Scholar
  3. 3.
    N. Bourbaki, Groupes et algèbres de Lie, Ch. IV, V, VI. Hermann: Paris, 1968.Google Scholar
  4. 4.
    C. Carré, “Le décodage de la régle de Littlewood-Richardson dans les triangles de Berenstein-Zelevinsky,” preprint, April 1991.Google Scholar
  5. 5.
    C. Davis, “Theory of positive linear dependence,” American Journal of Mathematics, vol. 76, pp. 733–746, 1954.Google Scholar
  6. 6.
    D. Gale, The theory of linear economic models, McGraw-Hill, New York, 1960.Google Scholar
  7. 7.
    I.M. Gelfand and A.V. Zelevinsky, “Polytopes in the pattern space and canonical basis in irreducible representations of g3,” Functional Analysis and Applications, vol. 19, no. 2, pp. 72–75, 1985 (in Russian).Google Scholar
  8. 8.
    I.M. Gelfand and A.V. Zelevinsky, “Multiplicities and regular bases for gn,” 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).Google Scholar
  9. 9.
    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.Google Scholar
  10. 10.
    I. Macdonald, Symmetric functions and Hall polynomials, Clarendon Press, Oxford, 1979.Google Scholar
  11. 11.
    E. Verlinde, “Fusion rule and modular transformation in 2d conformal field theory,” Nuclear Physics B, vol. 300, pp. 360–376, 1988.Google Scholar

Copyright information

© Kluwer Academic Publishers 1992

Authors and Affiliations

  • A.D. Berenstein
    • 1
  • A.V. Zelevinsky
    • 1
  1. 1.Department of MathematicsNortheastern UniversityBoston

Personalised recommendations