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# Annales de l’Institut Henri Poincaré D

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**Volume 5, Issue 3, 2018, pp. 339–386**

**DOI: 10.4171/AIHPD/57**

Published online: 2018-07-25

From orbital measures to Littlewood–Richardson coefficients and hive polytopes

Robert Coquereaux^{[1]}and Jean-Bernard Zuber

^{[2]}(1) Aix-Marseille Université, Marseille, France

(2) Université Pierre et Marie Curie Paris 6, France

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 multiplets of real eigenvalues for the associated Horn problem – are already known. In the integral case it can be thought of 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.

*Keywords: *Horn problem, honeycombs, polytopes, SU$(n)$ Littlewood–Richardson coefficients

Coquereaux Robert, Zuber Jean-Bernard: From orbital measures to Littlewood–Richardson coefficients and hive polytopes. *Ann. Inst. Henri Poincaré Comb. Phys. Interact.* 5 (2018), 339-386. doi: 10.4171/AIHPD/57