THE HONEYCOMB MODEL OF GLn(C) TENSOR PRODUCTS I: PROOF OF THE SATURATION CONJECTURE

Recently Klyachko [Kl] 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 are also sufficient, which was known as the saturation conjecture. In particular this proves Horn’s conjecture from 1962, giving a recursive system of inequalities [H]. Our principal tool is a new model of the Berenstein-Zelevinsky cone for computing Littlewood-Richardson coefficients [BZ, Ze], the honeycomb model. The saturation conjecture is a corollary of our main result, which is the existence of a particularly well-behaved honeycomb associated to regular triples (λ, μ, ν). 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? Another standard, if less symmetric, formulation of the problem above replaces Vν with its dual, and asks for which ν is V ∗ ν a constituent of Vλ⊗Vμ. In this formulation one can without essential loss of generality restrict to the case that λ, μ, and ν are polynomial representations, and rephrase the question in the language of Littlewood-Richardson coefficients; it asks for which triple of partitions λ, μ, ν is the Littlewood-Richardson coefficient c ∗ λμ positive. It is not hard to prove (as we will later in this introduction) that the set of such triples (λ, μ, ν) is closed under addition, so forms a monoid. In this paper we prove that this monoid is saturated, i.e. that for each triple of dominant weights (λ, μ, ν), (VNλ⊗VNμ⊗VNν) GLn(C) > 0 for some N > 0 =⇒ (Vλ⊗Vμ⊗Vν) GLn(C) > 0. This is of particular interest because Klyachko has recently given an answer to the general question above, which in one direction was only asymptotic [Kl]: Date: February 1, 2008. 1991 Mathematics Subject Classification. Primary 05E15, 22E46; Secondary 15A42. 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 [Kl], the inequalities are not independent – not all of them determine facets of the cone. This will be the subject of inquiry of our second paper [Hon2].