Talk:Parabolic subgroup of a reflection group
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Things not in the article[edit]
It is obvious (given the information that is in the article) that the relation "is a standard parabolic subgroup of" is transitive on Coxeter groups, and not obvious but not difficult to prove that the relation "is a parabolic subgroup of" is transitive for complex reflection groups. In the references I consulted, I was not able to find a clear statement of these transitivities at this level of generality: Kane asserts it (on page 58) only for finite real reflection groups.
The question "why parabolic?" is very natural. The correct answer for reflection groups is "because of the connection with algebraic groups". The correct answer for algebraic groups is ... complicated. There's excellent discussion in this MathOverflow thread about it, but it does not produce a conclusive answer and is not citable anyhow.
JBL (talk) 19:38, 10 January 2024 (UTC)
- I have added something about the name based on the MO thread. --JBL (talk) 22:00, 16 February 2024 (UTC)
Minor prose comment[edit]
This is not a big issue, but § Braid groups has a rather high density of parenthetical asides, enough to read a bit awkwardly to me. XOR'easter (talk) 22:27, 17 February 2024 (UTC)
- @XOR'easter: yes indeed, thanks -- a chronic problem when I write quickly. (The section was thrown together as a sort of placeholder -- I will definitely revist it.) ( <-- illustrating the problem ;) ). --JBL (talk) 17:39, 18 February 2024 (UTC)