From orbital measures to Littlewood-Richardson coefficients and hive polytopes

Abstract : The volume of the hive polytope (or polytope of honeycombs) associated with a Littlewood-Richardson coefficient of SU(n), or with a given admissible triple of highest weights, is expressed, in the generic case, in terms of the Fourier transform of a convolution product of orbital measures. Several properties of this function ---a function of three non-necessarily integral weights or of three real eigenvalues for the associated Horn problem--- are already known. In the integral case it can be thought as a semi-classical approximation of Littlewood-Richardson coefficients. We prove that it may be expressed as a local average of a finite number of such coefficients. We also relate this function to the Littlewood-Richardson polynomials (stretching polynomials) i.e., to the Ehrhart polynomials of the relevant hive polytopes. Several SU(n) examples, for n=2,3,...,6, are explicitly worked out.
Type de document :
Pré-publication, Document de travail
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Contributeur : Jean-Bernard Zuber <>
Soumis le : lundi 12 juin 2017 - 11:56:15
Dernière modification le : mercredi 14 juin 2017 - 01:11:50


  • HAL Id : hal-01536928, version 1
  • ARXIV : 1706.02793



R. Coquereaux, Jean-Bernard Zuber. From orbital measures to Littlewood-Richardson coefficients and hive polytopes. 2017. <hal-01536928>



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