Annales de l’Institut Henri Poincaré D
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Published online: 2018-07-25
From orbital measures to Littlewood–Richardson coefficients and hive polytopesRobert Coquereaux and Jean-Bernard Zuber (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