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 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 $(n)=2,3,\dots ,6$, are explicitly worked out.