Coefficients of positive roots in term of simple roots

Question

Let $\Phi$ be an irreducible root system and $\Phi^+$ be positive root system and $\Delta$ be base. For every positive root $\beta=\sum_{\alpha \in \Delta}m_\alpha\alpha$, the numbers $m_\alpha$ are called coefficients of $\beta$ when expressed in the simple roots. By a quick looking at lists of all irreducible root systems, I had an observation that always happens that $m_\alpha=1$ for some simple root $\alpha$ unless $\beta$ is either the highest root of $E_8$ or $F_4$. My question is that does there exist any proof for this statement?

Answer 1

I agree with Jyrki Lahtonen that your case by case check is a proof. I guess what you want is a better understanding. Here are some remarks. I suggest you search for "(co)minuscule (co)weights". The following exercices in Bourbaki [Lie] Ch. VI are relevant: 1.24, 2.5, 4.4, 4.15.

First, the case of the highest root. If $\alpha$ is such that $m_\alpha = 1$, the corresponding fundamental coweight $\varpi^\vee_\alpha$ is called minuscule. The minuscule coweights provide representatives for the nonzero elements of the quotient $P^\vee / Q^\vee$ (coweight lattice modulo coroot lattice). They can be read off the affine Dynkin diagram of $\Phi^\vee$: the minuscule coweights correspond to the nodes of the Dynkin diagram of $\Phi^\vee$ which are in the orbit of the affine node under the action of the automorphism group of the affine Dynkin diagram. So the exceptions $E_8$, $F_4$ and $G_2$ (you forgot the last one) correspond the related facts that the affine Dynkin diagrams for those types has trivial automorphism group, or that a quasisimple group of those types (I should say dual types but they are self-dual) is both adjoint and simply connected ($P = Q$...). They have no minuscule (co)weights.

Minuscule weights (for weights rather than coweights, exchange the roles of $\Phi$ and $\Phi^\vee$... that would have made the discussion easier) are important because many aspects of Lie theory become much easier (for example the corresponding fundamental Weyl module has weights in a single $W$-orbit... the Schubert calculus for the corresponding $G/P$ has only $0$ or $1$ coefficients...).

So in types other than $E_8$, $F_4$, $G_2$, the highest root always involves a $1$ (there are cominuscule weights). One can check that in those three types, there is a next biggest root, which does have a $1$ as coefficient (actually $\beta$ is a fundamental weight, $\varpi_8$, resp. $\varpi_1$, resp. $\varpi_2$, for a root $\alpha$ which has coefficient $2$; take $s_\alpha(\beta)$).

Now take a root $\gamma$ in any type, which is not one of the three exceptional cases (highest root in one of those three types). Consider its support (the subdiagram of the nodes corresponding to the simple roots involved; it is irreducible). We may replace $\Phi$ by the root system corresponding to this support. We are again in a non-exceptional case (those three types do not occur as strict subsystems of an irreducible root system). There is a simple root $\alpha$ appearing with multiplicity $1$ in the highest root, resp. the next-to-highest root. Then the coefficient of $\alpha$ in $\gamma$ is also $1$.