Talk:Parabolic subgroup of a reflection group
Parabolic subgroup of a reflection group is currently a Mathematics and mathematicians good article nominee. Nominated by JBL (talk) at 23:03, 1 September 2024 (UTC) Anyone who has not contributed significantly to (or nominated) this article may review it according to the good article criteria to decide whether or not to list it as a good article. To start the review process, click start review and save the page. (See here for the good article instructions.) Note: For purposes of criterion 1a ("understandable to an appropriately broad audience"): the high-quality sources that cover this topic are aimed at PhD students beginning to specialize or professional researchers in mathematics. In my opinion, this means that "an appropriately broad audience" should mean "advanced undergraduate mathematics majors and beginning PhD students in mathematics". |
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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)
Well-written article
[edit]I'd just like to congratulate you on an extremely well-written and readable article. I personally can't understand a single word of it, of course. But I can somehow tell that if I'd taken a class in group theory instead of sticking to analysis, I'd definitely be able to read this and understand what it said. :p
(Or maybe not. I hate discrete math. Who TF put all these holes between my numbers‽) – Closed Limelike Curves (talk) 00:49, 19 September 2024 (UTC)