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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.

Root system of $B_n$

Community
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Miguel
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2 Answers2

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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)

Stella Biderman
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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.

MonadBoy
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