# From orbital measures to Littlewood-Richardson coefficients and hive polytopes

## TL;DR

This paper links hive polytope volumes and Littlewood-Richardson coefficients through Fourier transforms of orbital measures, providing new formulas and explicit examples for SU(n).

## Contribution

It introduces a novel expression of hive polytope volumes as local averages of Littlewood-Richardson coefficients and relates these to Ehrhart polynomials.

## Key findings

- Expressed hive polytope volume as a local average of Littlewood-Richardson coefficients.
- Connected the Fourier transform of orbital measures to hive polytope volumes.
- Provided explicit SU(n) examples for n=2 to 6.

## 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.