On Simply Laced Lie Algebras and Their Minuscule Representations

The Lie algebra E6 may be defined as the algebra of endomorphisms of a 27-dimensional complex vector space MC which annihilate a particular cubic polynomial. This raises a natural question: what is this polynomial? If we choose a basis for MC consisting of weight vectors {Xw} (for some Cartan subalgebra of E6), then any invariant cubic polynomial must be a linear combination of monomials XwXw′Xw′′ where w + w′ + w′′ = 0. The problem is then to determine the coefficients of these monomials. Of course, the problem is not yet well-posed, since we still have a great deal of freedom to scale the basis vectors Xw. If we work over the integers instead of the complex numbers, then much of this freedom disappears. The Z-module M then decomposes as a direct sum of 27 weight spaces which are free Z-modules of rank 1. The generators of these weight spaces are well-defined up to a sign. Using a basis for M consisting of such generators, a little bit of thought shows that the invariant cubic polynomial may be written as a sum ∑