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Littlewood-Richardson semigroups

This note is an extended abstract of my talk at the workshop on Representation Theory and Symmetric Functions, MSRI, April 14, 1997. We discuss the problem of finding an explicit description of the semigroup $LR_r$ of triples of partitions of length $\leq r$ such that the corresponding Littlewood-Richardson coefficient is non-zero. After discussing the history of the problem and previously known results, we suggest a new approach based on the ``polyhedral'' combinatorial expressions for the Littlewood-Richardson coefficients.