Exercise 5, § 2

Assume that $R$ is irreducible, and choose a chamber $C$ of $R$ in $V$; denote by $B$ the corresponding basis of $R$.

a) Show that the minuscule weights (§ 1, Exerc. 24) of $R$ form a system of representatives in $P(R)$ of the non-zero elements of $P(R)/Q(R)$. (Apply to $R^-$ the corollary of Prop. 6.)

b) Let $X$ be a saturated subset of $Q(R)$ (§ 1, Exerc. 23). Assume that $X$ is non-empty and not reduced to $\{0\}$. Let $p$ be a non-zero element of $X$ of minimal length. Show that $p \in R$. (In view of a), $p$ does not satisfy condition (ii) of Exerc. 24 of § 1; hence, there exists $\alpha \in R$ such that $\langle p, \alpha^* \rangle \geq 2$, so p - α ∈ X. Since the length of p - α is strictly less than that of p, p - α = 0 so p ∈ R.)

c) Let p be a dominant weight not belonging to Q(R). Show that the saturated subset S(p) generated by p (cf. § 1, Exerc. 23) contains a unique minuscule weight. (Remark that S(p) is contained in a non-trivial class mod. Q(R); conclude by applying a) and Exerc. 24 c) of § 1.)

d) Let p be a dominant weight not belonging to Q(R). Prove that the following two properties are equivalent:

(i) p is minuscule.

(v) There is no dominant weight q ≠ p such that p - q is a linear combination of elements of B with non-negative integer coefficients.

(If q satisfies the conditions of (v), let p₁ be a minuscule weight belonging to S(q). Then, p₁ ≡ p mod. Q(R), p₁ < p, and a) shows that p is not minuscule. Hence, (i) ⇒ (v). Conversely, if p is not minuscule, let q ∈ S(p) - W.p; transforming q by an element of W if necessary, we can assume that q ∈ C̄; by Exerc. 23 of § 1, q < p, q ≠ p, and q ≡ p mod. Q(R). Hence, (v) ⇒ (i).)