Apiary Views of the Berenstein-zelevinsky Polytope, and Klyachko’s Saturation Conjecture

Recently Klyachko [K] has given linear inequalities on triples (λ, μ, ν) of dominant weights of GLn(C) necessary for the the corresponding Littlewood-Richardson coefficient dim(Vλ⊗Vμ⊗Vν) GLn(C) to be positive. We show that these conditions (and an evident congruency condition) are also sufficient, which was known as the saturation conjecture. In particular this proves Horn’s conjecture, giving a related system of inequalities [H, K]. 1. The saturation conjecture A very old and fundamental question about the representation theory of GLn(C) is the following: For which triples of dominant weights λ, μ, ν does the tensor product Vλ⊗Vμ⊗Vν of the irreducible representations with those high weights contain a GLn(C)invariant vector? One obvious condition is that λ+ μ+ ν be in the root lattice; otherwise there is no torusinvariant, much less GLn(C)-invariant, vector. Another standard formulation of the problem above is to ask for which ν does Vλ⊗Vμ contain the dual representation to Vν . Recently, Klyachko has given an answer to this question, which in one direction is only asymptotic: If Vλ⊗Vμ⊗Vν has a GLn(C)-invariant vector, then λ, μ, ν satisfy a certain system of linear inequalities derived from Schubert calculus. Conversely, if λ, μ, ν satisfy these inequalities, then there exists an integer N such that the tensor product VNλ⊗VNμ⊗VNν has a GLn(C)-invariant vector. Klyachko then states the saturation conjecture, that the full (nonasymptotic) converse of the first statement should be true (in presence of the obvious condition that λ+ μ+ ν be in the root lattice). He also points out an important consequence: Horn’s conjecture [H] from Date: March 2, 2008. Supported by an NSF Postdoctoral Fellowship. Partially supported by NSF grant DMS-9706764. Klyachko gives a finite set of inequalities, that as a set are necessary and sufficient for this asymptotic result. However, Chris Woodward has informed us that contrary to Klyachko’s unproven claim in [K], the inequalities are not independent – not all of them determine facets of the cone.