The number of extremal components of an extremal measure

TL;DR

This paper investigates the structure of extremal measures related to Littlewood-Richardson coefficients, revealing how the count of extremal components correlates with geometric data from the measure's support.

Contribution

It establishes a connection between the number of extremal components of an extremal measure and geometric information derived from its support.

Findings

Number of extremal components linked to geometric data

Unique decomposition of rigid measures into extremal parts

Provides a method to determine extremal count from support geometry

Abstract

It is known that the Littlewood-Richardson coefficients can be calculated using a certain class of measures, and these measures have a rigidity property when the coefficient is equal to 1. Rigid measures decompose uniquely into sums of extremal rigid measures. We show that the number of extremal summands is closely related with geometric data easily obtained from the support of the measure.