The saturation conjecture (after A. Knutson and T. Tao)

TL;DR

This paper provides a clear and comprehensive explanation of Knutson and Tao's proof of the saturation conjecture, demonstrating that the Littlewood-Richardson semigroup is saturated using the hive model for Berenstein-Zelevinsky polytopes.

Contribution

It offers a simplified and complete exposition of the proof of the saturation conjecture, highlighting the hive model's role and its equivalence to the original Littlewood-Richardson rule.

Findings

Proof confirms the saturation of the Littlewood-Richardson semigroup.

Hive model is equivalent to the original Littlewood-Richardson rule.

Provides a simplified exposition of the proof.

Abstract

In this exposition we give a simple and complete treatment of A. Knutson and T. Tao's recent proof (http://front.math.ucdavis.edu/math.RT/9807160) of the saturation conjecture, which asserts that the Littlewood-Richardson semigroup is saturated. The main tool is Knutson and Tao's hive model for Berenstein-Zelevinsky polytopes. In an appendix of W. Fulton it is shown that the hive model is equivalent to the original Littlewood-Richardson rule.