Tensor product varieties and crystals. ADE case

TL;DR

This paper introduces geometric varieties related to simple Lie algebras that model tensor products and multiplicities, establishing their connection to crystal bases and providing a geometric perspective on tensor product decompositions.

Contribution

It defines tensor product and multiplicity varieties associated with ADE Lie algebras and links their irreducible components to crystal bases and representation multiplicities.

Findings

Irreducible components form g-crystals isomorphic to canonical bases.

Number of components equals multiplicities in tensor products.

Provides geometric description of tensor product decompositions.

Abstract

Let g be a simple simply laced Lie algebra. In this paper two families of varieties associated to the Dynkin graph of 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 a structure of g-crystal isomorphic to the crystal of the canonical basis of the tensor product of several simple finite dimensional representations of 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 (on the level of crystals).