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This is an expanded version of the notes of my three lectures at a NATOAdvanced Study Institute ``Symmetric functions 2001: surveys of developmentsand perspectives" (Isaac Newton Institute for Mathematical Sciences, Cambridge,UK; June 25-July 6, 2001). Lecture I presents a unified expression due to A.Berenstein and the author for generalized Littlewood-Richardson coefficients (=tensor product multiplicities) for any complex semisimple Lie algebra. LectureII outlines a proof of this result; the main idea of the proof is to relate theLR-coefficients with canonical bases and total positivity. Lecture IIIintroduces cluster algebras, a new class of commutative algebras introduced byS. Fomin and the author in an attempt to create an algebraic framework forcanonical bases and total positivity.

From Littlewood-Richardson Coefficients to Cluster Algebras in Three Lectures