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Canonical bases for representations

Joel Kamnitzer (MIT - American Institute of Math)
January 25, 2006

A fundamental combinatorial question concerning Lie algebras is to calculate weight and tensor product multiplicities for their representations. The modern approach to this question is to construct special bases which are adapted to these calculations and then describe their combinatorics. There have been a number of constructions of such bases in the past 15 years, crystal bases by Kashiwara, canonical bases by Lusztig, the MV basis by Mirkovic-Vilonen, the basis given by components of quiver varieties by Nakajima, etc. Some of these constructions are more representation theoretic, while others more geometric, however all are quite non-trivial. We will survey some of these constructions and discuss the resulting combinatorics.