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.

32 pages, 4 figures. This version (V4): a few corrected typos

Cited by in corpus (7)

• On Horn's Problem and its Volume Function
• Harmonic analysis for rank-1 Randomised Horn Problems
• Dual formulations of Polyakov loop lattice models
• Revisiting Horn's Problem
• Derivative principles for invariant ensembles
• Projections of Orbital Measures and Quantum Marginal Problems
• Multiplicities, pictographs, and volumes