From Littlewood-richardson Coefficients to Cluster Algebras in Three Lectures

This is an expanded version of the notes of my three lectures at a NATO Advanced Study Institute “Symmetric functions 2001: surveys of developments and perspectives” (Isaac Newton Institute for Mathematical Sciences, Cambridge, UK; June 25 – July 6, 2001). Lecture I presents a unified expression from [4] for generalized Littlewood-Richardson coefficients (= tensor product multiplicities) for any complex semisimple Lie algebra. Lecture II outlines a proof of this result; the main idea of the proof is to relate the LR-coefficients with canonical bases and total positivity. Lecture III introduces cluster algebras, a new class of commutative algebras defined in [9] in an attempt to create an algebraic framework for canonical bases and total positivity.