Tensor product varieties and crystals: The \(ADE\) case. (English) Zbl 1048.20029

Malkin, Anton

Duke Math. J. 116, No. 3, 477-524 (2003).

Let \(L\) be a simple simply laced Lie algebra. The author describes tensor product and multiplicity varieties, associated to the Dynkin graph of \(L\), that are closely related to Nakajima’s quiver varieties. In particular, the author shows that the set of irreducible components of a tensor product variety can be equipped with the structure of an \(L\)-crystal isomorphic to the crystal of the canonical basis of the tensor product of several simple finite-dimensional representations of \(L\), and that the number of irreducible components of a multiplicity variety is equal to the multiplicity of a certain representation in the tensor product of several others. The decomposition of a tensor product into a direct sum is also described geometrically.

Reviewer: Melih Boral (Adana)

MSC:

20G05	Representation theory for linear algebraic groups
17B10	Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
14M15	Grassmannians, Schubert varieties, flag manifolds

Keywords:

tensor product varieties; crystals; multiplicity varieties

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