Duke Mathematical Journal

Tensor product varieties and crystals: The ADE case

Anton Malkin

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Abstract

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

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

Dates
First available in Project Euclid: 26 May 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1085598300

Digital Object Identifier
doi:10.1215/S0012-7094-03-11634-8

Mathematical Reviews number (MathSciNet)
MR1958096

Zentralblatt MATH identifier
1048.20029

Subjects
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}

Citation

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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