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,…, 6 , are explicitly worked out.

Locations

• Annales de l’Institut Henri Poincaré D Combinatorics Physics and their Interactions - View
• arXiv (Cornell University) - View - PDF
• HAL (Le Centre pour la Communication Scientifique Directe) - View - PDF
• DataCite API - View