I don't know what this symbol in root systems means (of coxeter groups)

Question

I'm reading Humphreys, Reflection groups and Coxeter groups. The section "Construction of root systems" and the books uses the symbol $ \mathop {\alpha}\limits^{\sim} $ to denote an special element. But I don't know what it is. I looked for it but there is nothing about it.

Answer 1

I think the answer lies in section 2.9, on page 40. I quote:

"/.../ (3) There is a natural partial ordering on $V$ /.../ When $\Phi$ is irreducible, there exists a unique highest root (a long root) relative to this ordering, denoted $\tilde{a}$; it plays a crucial role in 2.11 below as well as in Chapter 4/.../"

What the asker quotes is taken from section 2.10, so it would make sense if this definition from section 2.9 carried over.

Answer 2

I believe it is just a formal symbol that represents the element. They wanted something that was a modified $\alpha$ for continuity reasons, and decided to go with that symbol for unknown reasons. It's the same as if they had written $\alpha'$ (or even $x$ for that matter)