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.
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Annales de l’Institut Henri Poincaré (D) Combinatorics, Physics and their Interactions, European Mathematical Society, In press, pp.52 - 52
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  • ARXIV : 1706.02793

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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, European Mathematical Society, In press, pp.52 - 52. 〈hal-01536928〉

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