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To supplement what Pete says, I'd emphasize that "root systems" have been defined in a variety of ways for a variety of purposes related to Lie theory or to some type of "reflection" group. (For in …

After taking a closer look at the proof by Kac of Prop. 5.8 c), I can see that it's too sketchy to be followed easily. Here the generalized Cartan matrix is assumed to be of indefinite type, which I …

As Koen S points out, the longest element of an irreducible Weyl group is treated in an earlier question (in fact, it comes up in several questions). The question asked here presupposes a standard li …

This question (which I overlooked for a long time) reflects a natural notational confusion but is easy to answer. The Cartan integers themselves are unambiguous for each root system, but the meaning …

To compensate for my unfocused earlier comments it may be useful to supplement Richard's efficient answer based on Bourbaki's treatment of Coxeter elements in finite reflection groups. There is a sh …

As indicated in the comments, there is no need to redo the entire classification argument when passing from "semisimple" to "reductive". But it's useful to recall some of the history. The emphasis …

The role of the longest element in $W$ emerges only gradually in the Chevalley structure theory. This is developed similarly but in slightly different styles in the three books with the same title Li …

This is not a direct answer to your questions (which I still haven't understood completely from your formulation). But it seems important to place these questions within the extensive theoretical ba …

The classification of simple Lie algebras (over $\mathbb{C}$ or other sufficiently large field of characteristic 0) correlates these Lie algebras with the irreducible reduced root systems (in Bourbaki …

As my comment suggested, the question itself lacks detail and seems to be out of focus. For the somewhat intricate structure theory of a (real) simple or semisimple Lie group, there are standard tex …

The basic set-up here makes sense in the theory of abstract root systems if one brings (integral) weights into the picture, but it may be more natural to think about the classical characteristic 0 the …

Probably there is no explicit formula of the type you want. In any case, it's important to look first at the most accessible special cases (even though Weyl's formulas may be kept in the background) …

The problem with your highlighted formulation is that it's wrong as stated, unless for example you require that an "irreducible" representation be finite dimensional or have an integral highest weight …

The question is perhaps best answered not in the context of Lie theory but in the related setting of affine Weyl groups, where an irreducible root system in Bourbaki's sense leads to an extended Dynki …

I'm not sure whether there is an efficient way to answer your question (or a written reference), but it's possible to analyze the situation case-by-case. It's probably best to start with an irreducib …