The honeycomb model and regular rigid honeycombs

We introduce the {\em honeycomb} model of Berenstein-Zelevinsky polytopes. This associates a polytope of honeycombs to each triple of dominant weights of $GL_n$. The B-Z theorem says that the lattice points in these polytopes count Littlewood-Richardson numbers. Our main result is that for each regular triple of dominant weights, at least one vertex of this polytope is a particularly nice honeycomb. One corollary of this is Klyachko's saturation conjecture, which says that the semigroup of ({\em not} necessarily regular) triples with $LR>1$ is saturated -- the nice honeycombs are necessarily integral. In this talk we hope to get to another corollary, a description of those triples of regular dominant weights for which the Littlewood-Richardson coefficient is exactly 1. This work is joint with Terry Tao of UCLA.