Journal of Algebraic Combinatorics
, Volume 1, Issue 1, pp 722
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.
 Title
 Triple Multiplicities for sl(r + 1) and the Spectrum of the Exterior Algebra of the Adjoint Representation
 Journal

Journal of Algebraic Combinatorics
Volume 1, Issue 1 , pp 722
 Cover Date
 199205
 DOI
 10.1023/A:1022429213282
 Print ISSN
 09259899
 Online ISSN
 15729192
 Publisher
 Kluwer Academic PublishersPlenum Publishers
 Additional Links
 Topics
 Keywords

 tensor product multiplicities
 systems of linear inequalities
 Authors

 A.D. Berenstein ^{(1)}
 A.V. Zelevinsky ^{(1)}
 Author Affiliations

 1. Department of Mathematics, Northeastern University, Boston, MA, 02115