Journal of Algebraic Combinatorics

, Volume 1, Issue 1, pp 7-22

First online:

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

  • A.D. BerensteinAffiliated withDepartment of Mathematics, Northeastern University
  • , A.V. ZelevinskyAffiliated withDepartment of Mathematics, Northeastern University

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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