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Article Dans Une Revue Annales de l’Institut Henri Poincaré (D) Combinatorics, Physics and their Interactions Année : 2018

From orbital measures to Littlewood-Richardson coefficients and hive polytopes

Résumé

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.
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Dates et versions

hal-01536928 , version 1 (24-04-2018)

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Citer

R. Coquereaux, Jean-Bernard Zuber. From orbital measures to Littlewood-Richardson coefficients and hive polytopes. Annales de l’Institut Henri Poincaré (D) Combinatorics, Physics and their Interactions, 2018, Combinatorics, Physics and their Interactions, 5 (3), pp.339-386. ⟨10.4171/AIHPD/57⟩. ⟨hal-01536928⟩
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