Let $\Phi$ be an irreducible root system and $\Delta$ a simple system (base). Let $W$ be the Weyl group of $\Phi$. Let $\theta$ be the highest root and $h^\vee$ be the dual Coxeter number. Choose the shortest $w \in W$ such that $w(\alpha_i)=\theta$ for some simple root $\alpha_i$. I got the result that the length of $w$ can be determined by $l(w) = h^\vee −2$ (for instance see Lemma 5.1 in here). However I'm not given yet any explicit proof for that result. Can anyone give me proof or references where this formula is introduced and proved?
Thanks in advance.
Best Answer
Apparently this isn't discussed in any of the published literature, even in the numerous exercises for Bourbaki's Chapter VI on root systems in Lie Groups and Lie Algebras. I'm not sure how strong the evidence is for the assertion here that the minimal length is always $h^\vee -2$. Even though it's true for some small rank cases, has it actually been verified for all types? For example, in the respective types $E_6, E_7, E_8$ we have $h = h^\vee = 12, 18, 30$, and in type $F_4$ we have $h=12$ but $h^\vee = 9$.
The underlying question here is whether an element $w$ of minimal length taking a (necessarily long) simple root to $\theta$ can be characterized uniformly in some way. (A further question is how interesting such a result would be: are there are useful consequences if $\ell(w)$ does turn out as predicted?) From a look at small rank examples, I don't yet see any uniform pattern here for the choice of $w$. So it's important to clarify how far the assertion about $\ell(w)$ has been verified case-by-case. Probably the length question might be studied more uniformly in two ways: using just the axiomatics of root systems, or using specific features of the adjoint representation of a simple Lie algebra. But neither approach looks straightforward.
[By the way, the dual Coxeter number has come up in a number of questions on MO. The geometric work by Coxeter treated arbitrary finite real reflection groups, but later it was seen that the order $h$ of a Coxeter element in a crystallographic group (such as the Weyl group of an irreducible root system) can be characterized as 1 plus the height of the highest root. Then it was useful in Kac-Moody theory to define $h^\vee$ to be 1 plus the height of the highest short root if there are two root lengths.]
UPDATE: As the comments below indicate, the length in question is treated by Cellini and Papi (Prop. 7.1 in their 2004 paper in Advances in Math. which I provided a link for) as well as in the 2011 thesis of a student of Papi. I had overlooked the paper, so I was overly pessimistic about finding an explicit discussion in the published literature.