From Littlewood-Richardson coefficients to cluster algebras in three lectures

Details From Littlewood-Richardson coefficients to cluster algebras in three lectures From Littlewood-Richardson coefficients to cluster algebras in three lectures

2001-12-06

arxiv.org/abs/math/0112062v2

Mathematics

Representation Theory

Latex, 17 pages, Theorem 3.2 corrected

Scientific paper

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 due to A. Berenstein and the author 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 introduced by S. Fomin and the author in an attempt to create an algebraic framework for canonical bases and total positivity.

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