Duke Mathematical Journal

Tensor product varieties and crystals: The ADE case

Anton Malkin

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Let $\mathfrak {g}$ be a simple simply laced Lie algebra. In this paper two families of varieties associated to the Dynkin graph of $\mathfrak {g}$ are described: tensor product and multiplicity varieties. These varieties are closely related to Nakajima's quiver varieties and should play an important role in the geometric constructions of tensor products and intertwining operators. In particular, it is shown that the set of irreducible components of a tensor product variety can be equipped with the structure of a $\mathfrak {g}$-crystal isomorphic to the crystal of the canonical basis of the tensor product of several simple finitedimensional representations of $\mathfrak {g}$, 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. Moreover, the decomposition of a tensor product into a direct sum is described geometrically.

Article information

Duke Math. J., Volume 116, Number 3 (2003), 477-524.

First available in Project Euclid: 26 May 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 17Bxx: Lie algebras and Lie superalgebras {For Lie groups, see 22Exx}
Secondary: 20Gxx: Linear algebraic groups and related topics {For arithmetic theory, see 11E57, 11H56; for geometric theory, see 14Lxx, 22Exx; for other methods in representation theory, see 15A30, 22E45, 22E46, 22E47, 22E50, 22E55}


Malkin, Anton. Tensor product varieties and crystals: The ADE case. Duke Math. J. 116 (2003), no. 3, 477--524. doi:10.1215/S0012-7094-03-11634-8. https://projecteuclid.org/euclid.dmj/1085598300

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