### A positive proof of the Littlewood-Richardson rule using the octahedron recurrence.

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The set of conjugacy classes appearing in a product of conjugacy classes in a compact, $1$-connected Lie group $K$ can be identified with a convex polytope in the Weyl alcove. In this paper we identify linear inequalities defining this polytope. Each inequality corresponds to a non-vanishing Gromov-Witten invariant for a generalized flag variety $G/P$, where $G$ is the complexification of $K$ and $P$ is a maximal parabolic subgroup. This generalizes the results for $SU\left(n\right)$ of Agnihotri and the second author and Belkale on...

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