Equations determining the orbit of the highest weight vector in the adjoint representation

摘要

We explicitly construct a set of quadratic equations defining the highestweight vector orbit for adjoint representations of Chevalley groups of typesD_l, E_6, E_7, and E_8. The combinatorics of these equations is related to thecombinatorics of embeddings of the root system of type A_3. We believe that theconstructed equations provide a prominent framework for calculations withexceptional groups in adjoint representations, which is particularlyinteresting for groups of type E_8.