Combinatorics of Minuscule Representations

Let g be a simple Lie algebra over ℂ, corresponding to an n × n Cartan matrix, and let h be the subalgebra of g defined in Section 4.3. A Chevalley basis for g is by definition a basis of g of the form

{hi : 1 ≤ i ≤ n} ∪{xα : α ∈ Φ},

where Φ is the set of roots of g, satisfying the conditions

[h, xα] = α(h)xα for all h ∈ h and α ∈ Φ,

[xα, x–α] = –h,

[xα, xβ] = 0 if α, β ∈ Φ, α + β ∉ Φ and α + β ≠ 0,

[xα, xα] = Nα,βxα+β if α, β ∈ Φ, where

Nα,β = N–α,–β = ±(p + 1),

where p ≥ 0 is the greatest integer such that α. pβ ∈ Φ, and hα and h are as in Section 4.3.

It turns out that the choice of signs above is somewhat delicate, but as we shall show in Chapter 7, the theory of full heaps can be used to construct explicit Chevalley bases for simple Lie algebras over ℂ, except in types E8, F4 andG2, where no corresponding full heap exists.

Kac's asymmetry function

In Section 7.1, we assume that the Weyl group W is associated with a simply laced Cartan matrix; that is, a generalized Cartan matrix A of finite type. Recall from Theorem 4.2.3 (iii) and (iv) that, in this case, all roots have the same length.