The Saturation Conjecture ( After

The purpose of this exposition 1 is to give a simple and complete treatment of Knutson and Tao's recent proof of the saturation conjecture [8]. If λ is a partition of length at most n, let V λ denote the corresponding highest weight representation of GL n (C). Define This set is important in numerous areas besides representation theory. In Schubert calculus it describes when an intersection of Schubert cells must be non-empty. In combinatorics, a triple is in T n if and only if there exists a Littlewood-Richardson skew tableau with shape ν/λ and content µ. It is well known that T n ⊂ Z 3n is a semi-group under addition, a fact which Zelevinsky attributes to Brion and Knop [10]. Klyachko has given [7] a nice description of the saturation A triple (λ, µ, ν) is in ¯ T n if and only if the entries of λ, µ, and ν satisfy certain inequalities that come from Schubert calculus (see also [5]). This made the following conjecture particularly important. In other words T n is saturated in Z 3n. Note that the implication ⇒ is a trivial consequence of the fact that T n is a semi-group or of the original Littlewood-Richardson rule. In July 1998, Knutson and Tao gave a proof of this conjecture, using two wonderful new descriptions of Berenstein-Zelevinsky polytopes called the honeycomb and hive models [8]. The goal of this exposition is to present a simple and complete proof based only on the hive model. Since the final version of Knutson and Tao's paper will likely be based on honeycombs alone, we hope that this may be useful. Most constructions used here come directly from the first version of Knutson and Tao's preprint, even if they may be replaced by honeycomb equivalents in their published paper. One innovation, in Section 3, is the construction of a graph from a hive, which is used to simplify their argument. In an appendix of Fulton it is shown that the hive model is equivalent to the original Littlewood-Richardson rule.