Talk:Parabolic subgroup of a reflection group

Things not in the article

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

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

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)

Sourcing issue

Hi @JayBeeEll: this is effectively the same thing as a listserv; we wouldn't consider professors emailing back and forth about research questions to be reliable, so we shouldn't consider them doing the same thing online for all to see reliable either. voorts (talk/contributions) 00:29, 28 October 2024 (UTC)

Hi voorts, sorry for the delayed response. At the level of what WP:GUNREL says, it's extremely clear: The source may still be used for uncontroversial self-descriptions, and self-published or user-generated content authored by established subject-matter experts is also acceptable. The linked page is a discussion between a number of subject-matter experts, in a scholarly (although unrefereed) venue; it definitely qualifies under the second half of the sentence I've quoted. (This description might also apply to some listservs, but that seems neither here nor there.) Personally I think the discussion there provides some small but very clear added value beyond what is found in Borel -- namely, the commentary of James E. Humphreys on what is found in Borel -- and that a reader interested in understanding this name is best served by being given both citations (even though there is no piece of information in the article here that relies on the MO thread). If you do not find this compelling, perhaps we can solicit a third opinion from WT:WPM? --JBL (talk) 00:28, 31 October 2024 (UTC)

No worries regarding the delay, and thank you for the response. I think that both references aren't needed per WP:BESTSOURCES and WP:TIERS, but your position on GUNREL is reasonable, so I'm fine with maintaining the status quo. Good luck with the GA nom. Best, voorts (talk/contributions) 02:17, 31 October 2024 (UTC)

I think you (plus the realization that I didn't include any content from the link, presumably because I was skeptical of reliability) have convinced me better out than in; for the record I preserve the citation here:

Chow, Timothy; et al. (2010), "Why are parabolic subgroups called "parabolic subgroups"?", MathOverflow, retrieved 2024-02-16

Thanks, JBL (talk) 00:36, 1 November 2024 (UTC)

Coxeter groups vs. systems

Rather than referring to standard parabolic subgroups of a Coxeter group W with a finite set S of simple reflections, I think it would be clearer to define standard parabolic subgroups of a Coxeter system $(W,S)$.

To quote Björner & Brenti (2005) p.2: When referring to an abstract group as a Coxeter group, one usually has in mind not only W but the pair $(W,S)$, with a specific generating set S tacitly understood. Some caution is necessary in such cases, since the isomorphism type of $(W,S)$ is not determined by the group W alone.

Our Coxeter groups article gives a concrete example: the Coxeter groups of type $B_{3}$ and $A_{1}\times A_{3}$ are isomorphic but the Coxeter systems are not equivalent, since the former has 3 generators and the latter has 1 + 3 = 4 generators.

This is particularly relevant because we have an example referring to the hyperoctahedral group $S_{n}^{B}$. When n = 3, its group structure is insufficient to identify its standard parabolic subgroups: interpreting the group as the Coxeter system $B_{3}$ yields a Boolean lattice of 2^3 = 8 parabolic subgroups, whereas interpreting it as the Coxeter system $A_{1}\times A_{3}$ yields a Boolean lattice of 2^4 = 16 parabolic subgroups.

For clarity, it might be preferable to adopt a similar approach to Coxeter complex, referring to Coxeter groups (for simplicity) in the article lede, then switching to Coxeter systems (for precision) in the article body, from the Background section onwards.

While looking into this, I noticed a second issue with our hyperoctahedral example: we claim that the hyperoctahedral group (symmetric with respect to swaps between unsigned labels $\{1,\ldots ,n\}$) has maximal standard parabolic subgroups the stabilizers of $\{i+1,\ldots ,n\}$ for $i\in \{1,\ldots ,n\}$ (not symmetric with respect to these swaps). I went back to the source, Björner & Brenti (2005) p.246, and they're using a presentation of the hyperoctahedral group as a Coxeter system which is not invariant to unsigned label swaps. I think we need to add this specific presentation (set of generators) to the example for it to be correctly specified.

Lastly (sorry about all the nitpicks), the solution to this example is stated to be the stabilizer of a particular set. WP does not (currently) define this anywhere: stabilizer subgroup redirects to Group action#Fixed points and stabilizer subgroups, which defines the stabilizer subgroup $\operatorname {Stab} _{G}(x)$ of a point x, but does not define what the stabilizer subgroup $\operatorname {Stab} _{G}(X)$ of a set X should be. https://math.stackexchange.com/questions/2109769/definition-of-stabilizer-of-a-set claims there are two definitions in common use. In fact, Björner & Brenti (2005) p.307 uses a third definition, which AoPS refers to as the strict stabilizer. (AFAICT, this should be identical to the more common non-pointwise definition for finite sets, but may disagree for infinite sets.) So yeah ... I think we also need to write out what $\operatorname {Stab} _{S_{n}^{B}}(\{i+1,\ldots ,n\})$ means to clarify which definition is being used here. Preimage (talk) 13:14, 6 May 2025 (UTC)

I appreciate your giving the article a careful once-over. I am currently knee-deep in final exam grading, so I don't have time to fully engage with everything you've written, but, quickly: (1) yes for standard parabolic subgroups they are relative to the Coxeter system not just the group, and your suggestion of being more careful about this outside the lead is good; (2) I don't understand your comment about the hyperoctahedral group ("While looking into this ..."); and (3) the answer to the MSE question you linked gives the correct resolution here (no one being careful says "stabilizer" when they mean "pointwise stabilizer" rather than "setwise stabilizer"), and you are correct that when considering group actions on finite sets it is equivalent to ask for $g(A)=A$ and $g(A)\subseteq A$, so that for group actions on finite sets there is really only one unambiguous concept (which, as the MSE answer points out, coincides with the stabilizer of a single point when we extend the group action to the powerset in the usual way), but it is also annoying that these standard conventions are not clearly written at Group action; I am a little skeptical that this belongs in the article body here, but perhaps an efn in the relevant spot? --JBL (talk) 20:27, 6 May 2025 (UTC)

Thanks. I'm new to this area, so don't assume everything I say makes sense.

I think you're right re: (3). Upon rereading, the AoPS link is more general, defining the stabilizer monoid and strict stabilizer monoid of a monoid action. For group actions, these are equivalent for finite sets, but only the latter is guaranteed to be a group for infinite sets, implying the stabilizer subgroup must be defined using the strict stabilizer criterion. I'm just a bit wary because the MSE link includes the comment unfortunately (?) one of the most widely read and most influential books on permutation groups, the one by Wielandt, uses the ... pointwise stabilizer, and the current description in Group action#Fixed points and stabilizer subgroups conflates isotropy groups (which use the "pointwise stabilizer" definition) with stabilizer subgroups (they're equivalent for points, but not for larger sets). But provided we can find sources to confirm there's only one standard convention, we should define the stabilizer subgroup of a set in Group action, rather than cluttering up this article. (We're both busy right now, this can probably wait until later if we decide to do it at all.)

Re: (2), could we say something like the following?

• The Coxeter system $B_{n}$ may be realized as the hyperoctahedral group $S_{n}^{B}$, which consists of all signed permutations of $\{\pm 1,\ldots ,\pm n\}$ (that is, the bijections w on that set such that $w(-i)=-w(i)$ for all i), paired with generators $(1\ -1)$ and $(1\ 2)(-1\ -2)$, ..., $(n-1\ n)(-n+1\ -n)$. Its maximal standard parabolic subgroups are the stabilizers of $\{i,\ldots ,n\}$ for $i\in \{1,\ldots ,n\}$.

Lastly, I think the image examples are really nice, but I think we could flesh them out a bit more:

• For example, $S_{2}^{B}$ has standard parabolic subgroups $\{\iota ,(1\ -1)\}$ and $\{\iota ,(1\ 2)(-1\ -2)\}$, which under conjugation yield the additional parabolic subgroups $\{\iota ,(2\ -2)\}$ and $\{\iota ,(1\ -2)(-1\ 2)\}$. These form a non-Boolean lattice, as depicted in Figure 3.
• [Then mention that $S_{2}^{B}$ is isomorphic to the dihedral group $D_{2\times 4}$ as Coxeter systems, explaining why we get the same lattice of parabolic subgroups of the dihedral group $D_{2\times 4}$ in the previous figure.]

Preimage (talk) 03:04, 7 May 2025 (UTC)

@Preimage: I have made some edits following your suggestion about Coxeter system versus Coxeter group. There are a few places where I have left the construction "standard parabolic subgroup of [group symbol]" in which the set S has just been specified but not named, on the theory that no realistic confusion is possible. Please let me know if you think I've missed some instances (or, of course, feel free to fix them yourself!). I also added the word "setwise" before "stabilizer" in the section on the hyperoctahedral group. I am still digesting your other suggestion about that section. --JBL (talk) 17:46, 13 May 2025 (UTC)

Ok I like your other suggestions, too -- I have implemented them (not quite verbatim). It's always hard to source specific concrete examples like this, but let me not worry about that right now. --JBL (talk) 18:01, 13 May 2025 (UTC)

This is too complicated

@JayBeeEll Every so often, I take a trip through WP:GAN to see what makes sense for me to review. This article has been on the list for 9 months now and nobody's picked it up. The note says "an appropriately broad audience" should mean "advanced undergraduate mathematics majors and beginning PhD students in mathematics" but this is at odds with the message of WP:TECHNICAL and suspect that's the reason why nobody's touched it yet.

I can't get through the lead. I can't even get through the first two sentences. I'm not a mathematician, but as somebody trained in engineering and computer science, I have a better grounding in math than most of our readers. I feel like I should be able to at least get through the first paragraph without my eyes glazing over. I note that @Closed Limelike Curves, who also professes to have a math backgroud, had a similar reaction, i.e. I personally can't understand a single word of it.

Given your note, I didn't think it was fair for me to put this in a formal GA review, but I do think it needs to be said. I encourage you to find a way to make this more approachable to an audience less advanced than a PhD student. At the very least, I would hope a lead could be written which gives a more mainstream reader a basic understanding of the topic before getting into the deep dive. RoySmith (talk) 14:19, 1 June 2025 (UTC)

Even being part of the target audience for this article (as someone who has taken undergraduate level abstract algebra), I find the first two sentences to be too vague to even be considered a definition. Calling them a special kind of subgroup does not help me understand what these subgroups actually are or what we are studying. Compare this to the Normal subgroup article that defines them as a subgroup that is invariant under conjugation by members of the group of which it is a part. Telling the reader that The precise definition of which subgroups are parabolic depends on context is a bit of a cop-out to giving an actual definition of the groups and makes it harder for the reader to comprehend what the subject is about, contributing to being technical. I think the article should stick to defining the subgroups first in one of the given contexts, and then later in the lead mention the other context(s) relevant to these groups. Giving the definition in the finite case, which the lead says is where they coincide (without a citation directly in the lead) would help the reader get an immediate grasp of the concept. Gramix13 (talk) 04:08, 2 June 2025 (UTC)

Looking at the lede, it occurred to me that maybe it would help to focus on the Coxeter case there, and work in material about complex reflection groups later. Or even to funnel in to the Coxter case by introducing the example of S_n, where we're just looking at subgroups generated by sets of adjacent transpositions. Russ Woodroofe (talk) 19:11, 2 June 2025 (UTC)

Thanks, Russ, the idea of starting with a concrete example is a wonderful suggestion! I'm in the middle of something else but I will think about implementation and try to put something together later this week. --JBL (talk) 19:25, 2 June 2025 (UTC)

@Russ Woodroofe, Jacobolus, Gramix13, Tito Omburo, and Brirush: At some point you have made some constructive comment about the article (for which thank you, even if I may have been grumpy about them earlier). I've recently rewritten the lead section. If you had any further concrete comments about it (or just see some easy way to improve it yourself), that would be welcome. (And if not, then of course no worries, and apologies for pestering you!) --JBL (talk) 20:45, 16 June 2025 (UTC)

Missed @Preimage: (and of course the sadly departed @XOR'easter: :(. --JBL (talk) 20:47, 16 June 2025 (UTC)

Thank you for the ping (I don't find it pestering in the slightest)!

The leading paragraph is much more accessible now, although I have three suggestions that I think might help. Firstly, I actually find the footnote to be a much better visual for me to understand the concept of a standard parabolic subgroup in the context of the symmetric group (I could see an image being made in the future illustrating those subgroups or permutations in those subgroups), and so I think this should be swapped with the text explaining how the subgroups are just generated by subsets of $S$, with this description then following outside of a footnote to lead into the second paragraph discussing Coxeter groups.

My second suggestion was to add a brief note of what a conjugate of a set is, I figure the reminder might be helpful for the (undergraduate) reader to remember. I don't feel too strongly about this note, so feel free to disregard if you feel inclinded.

My third suggestion would be to include concrete examples of these parabolic subgroups look like for the symmetric group in the lead to help the reader better grasp what they might look like in practice and out of abstraction. Maybe giving non-examples of those subgroups (if they do exist) might also be illustrative if possible.

The second paragraph does click with me better now with the new leading paragraph, and I feel it gives a more satisfying general definition of what a parabolic subgroup of a reflection group really is, even if I don't quite know what the generators of the subgroup are or what subsets the subgroups are attempting to fix.

The last two paragraph could be combined as they are short on their own, or they could be expanded to not be as short.

Overall I think this is a great improvement in the delivery of the lead to this article. Gramix13 (talk) 21:34, 16 June 2025 (UTC)

GA review

This review is transcluded from Talk:Parabolic subgroup of a reflection group/GA1. The edit link for this section can be used to add comments to the review.

Nominator: JayBeeEll (talk · contribs) 23:03, 1 September 2024 (UTC)

Reviewer: Kingsif (talk · contribs) 21:20, 7 July 2025 (UTC)

Hi, I'm Kingsif, and I'll be doing this review. This is an automated message that helps keep the bot updating the nominated article's talkpage working and allows me to say hi. Feel free to reach out and, if you think the review has gone well, I have some open GA nominations that you could (but are under no obligation to) look at. Kingsif (talk) 21:20, 7 July 2025 (UTC)

Hello @Kingsif, I wanted to ask what the progress on this review is as its been a week since the start of the review and I haven't seen any other comments yet? I know you mentioned on JBL's talk page that you've had work taking up time away from editing, and that you've acknowledge the difficulty in reviewing this article, so no pressure if you need more time for doing this review. I just want to know where things stand right now as someone watching this review. And also, thank you for offering to take on this review in the first place considering the circumstances of how long this article has gone unreviewed and lingered in nominations. Gramix13 (talk) 16:27, 14 July 2025 (UTC)

Yeah, based on discussion with JBL that they're also quite busy this month, I've been taking this review criteria point by criteria point so far. I can add some comments but (for JBL) no pressure to address anything in a rushed way. Kingsif (talk) 20:12, 14 July 2025 (UTC)

Thanks -- I am travelling through the end of this week but I'm looking forward to digging in next week! JBL (talk) 00:41, 23 July 2025 (UTC)

Good Article review progress box

Criteria: 1a. prose () 1b. MoS () 2a. ref layout () 2b. cites WP:RS () 2c. no WP:OR () 2d. no WP:CV () 3a. broadness () 3b. focus () 4. neutral () 5. stable () 6a. free or tagged images () 6b. pics relevant () Note: this represents where the article stands relative to the Good Article criteria. Criteria marked are unassessed

Source spot-check

This spot-check will cover six of the citations in the article. Some of the sources are accessible to me on web, I'll ask if there's anything not accessible I would like to check.

• Ref #2 (Kane 2001, pp.8–14): I recognise that the selected pages from the source are describing a mathematical theory that has been summarised in the article. Going over it, the text, especially these dihedral structures are indicative of the general situation, confirms the article's summary. I believe the summary is appropriate. Y
• Ref #6 (Lehrer & Taylor 2009, p.1): Would you be able to explain the content of this page in the source a bit? I can see how it implies or mentions the complexification of real reflections, but not the rest of the sentence in the article ("Every real reflection group can be complexified to give a complex reflection group, so the complex reflection groups form another generalization of finite real reflection groups.") I appreciate there is also another ref at the end of this sentence.
• Ref #12 (Björner & Brenti 2005, p.17): Source is a selection of proofs with different variables, showing that the overall statement in footnote e is true. Y
• Ref #32 (Gobet 2017): Source is a proof example which, especially X is not conjugate to any of the three rank 2 standard parabolic subgroups of W, seems to confirm that article sentence. Y
• Ref #37 (Borel 2001): Note that the page number that I get for this is 121. Source confirms info in footnote j. The source also suggests the term is anachronistic, which could be mentioned in said footnote? Y
• Ref #38 (Digne & Michel 1991, pp.19–21): Source's 1.4 through 1.6 confirms the info in the article section. This is the only source for the section, is that it? Y

I'm not sure exactly where to put my responses, hopefully here is good! About the second point (Ref #6): The sentence "In 1951, ..." in the reference verifies the sentence "Especially, if one replaces ..." in the article. The phrase "These groups include the Euclidean groups" verifies that the complex groups generalize the real groups. Although the word "complexification" appears, it does not quite verify the assertion in the article that "Every real reflection group can be complexified to give a complex reflection group"; however, that statement is verified directly by the other footnote (Ref #7), which includes the sentence "Among these [complex reflection] groups are the [finite real] reflection groups we have been studying (complexified)." In fact I think Ref #6 is superfluous here and Ref #7 covers both sentences. --JBL (talk) 19:44, 29 July 2025 (UTC)

Criteria

• 1a: understandable to a sufficiently broad audience; prose and grammar correct and suitable.
  • As discussed, this is a topic which is introduced at undergraduate level mathematics. The applicable broad audience is therefore this group of people. Based on comments at the article's talkpage, especially from Gramix13, and subsequent edits in the last month, the article is written in a way to be understood by this group.
    • To expand on this, JayBeeEll's exact comment was "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"." I take this to mean that the concept would be introduced at the academic level of an undergraduate, with deeper and independent understanding at academic level of a postgraduate researcher. One would assume the readers looking for Wikipedia's summary are more likely at the lower end of this range - and being understandable to an undergraduate does not mean limiting the content to this level. Effectively, we don't want an article that can only be understood by the people it's citing, as that's not useful to anyone. Kingsif (talk) 11:38, 20 July 2025 (UTC)

      To be precise: I don't think the topic of this article is introduced to undergraduates, with perhaps exceptions for advanced undergraduates doing research projects (via an REU or whatever), and the literature that discusses it is almost universally aimed at graduate students or professional mathematicians. However I think the group of people who could conceivably understand at least some part of the article includes sufficiently advanced undergraduates (say those who have taken a good course in abstract algebra), and so my target audience (per WP:ONEDOWN) includes them. --JBL (talk) 17:22, 1 August 2025 (UTC)

  • I have found no prose errors yet, and I find the article does a good job combining the mathematical equations; summary prose; and descriptive prose.

  Kingsif (talk) 22:12, 19 July 2025 (UTC)

  Thanks! JBL (talk) 17:17, 1 August 2025 (UTC)

  I must confess that the comments on the talk page were specifically for the lead, which I would agree would meet the criteria for being understandable to an audience of undergraduate mathematics students (although this is certainly not introduced at that level at least in my own experience). I can't say whether or not the rest of the article meets being understandable since I never really took the time to continue thoroughly reading it over, but I do think the rewrite in the lead was promising in this respect. Gramix13 (talk) 22:21, 19 July 2025 (UTC)

Noted. If you have the time and inclination, it'd be helpful to have the eyes on the rest of the article; I was otherwise going to ask at the Mathematics WikiProject. Kingsif (talk) 22:28, 19 July 2025 (UTC)

Sure, I can go ahead and take this as an opportunity to read through the rest of the article to see if its understandable. I'll leave comments here if I find anything that might be unclear or could be explained better for clarity. Gramix13 (talk) 22:35, 19 July 2025 (UTC)

Thanks, that'd be really helpful. Kingsif (talk) 22:43, 19 July 2025 (UTC)

I don't see the Bruhat order being explained when it is introduced as an invariant on standard parabolic subgroups, although this is the only time the article mentions it. A brief description of what it means would help since this could be the first article for someone learning about Coxeter groups and is therefore unaware of what that ordering is. Gramix13 (talk) 23:09, 19 July 2025 (UTC)

Good point. I think that, unlike the length function, getting into an actual definition of Bruhat order would be way too far afield (and probably not too helpful), so I've just added a quick contextual gloss . --JBL (talk) 17:32, 1 August 2025 (UTC)

That looks good to me, and probably a good call not to go too much into the weeds unnecessarily. Gramix13 (talk) 19:11, 1 August 2025 (UTC)

Consequently, the lattice of standard parabolic subgroups of W is a Boolean lattice Is this because the lattice is isomorphic to the subsets of the generators? If so, that being communicated could give the reader a much better picture in their mind of how to view the standard parabolic subgroups simply as analogous to the subsets of the generator, including with the operations of intersection and union/generated subgroup. It's possible the reader might not immediately see why it should be a Boolean lattice (or even know what that means), but certainly they will be familiar with the structure of subsets which might be more approachable to an undergraduate. Gramix13 (talk) 23:14, 19 July 2025 (UTC)

Is this because the lattice is isomorphic to the subsets of the generators? Yes, good, thank you. I've spelled it out more so you can get the idea without knowing the jargon: . --JBL (talk) 17:28, 1 August 2025 (UTC)

This looks perfect! Gramix13 (talk) 19:12, 1 August 2025 (UTC)

In terms of the Coxeter–Dynkin diagram... I think it would be helpful to also explain what this kind of diagram is since I don't think its something all undergraduates would know. Gramix13 (talk) 23:21, 19 July 2025 (UTC)

Glossed: . --JBL (talk) 18:09, 1 August 2025 (UTC)

That seems acceptable to not go too much into those diagrams. I did correct an error in the math tags I saw there, might be good to double check it. Gramix13 (talk) 19:16, 1 August 2025 (UTC)

Yes, thank you, obviously I need to proofread my edits! --JBL (talk) 19:19, 1 August 2025 (UTC)

The collection of all intersections of subsets of these hyperplanes... This sounds vague, what subsets of hyperplanes are we intersecting by? All of them? Just the hyperplanes? Gramix13 (talk) 23:28, 19 July 2025 (UTC)

So what I mean is, you have all these hyperplanes; take any subset of these hyperplanes and take the intersection of the planes in that subset to get a subspace. Now do that in all possible ways. That's the collection I'm talking about here. I agree that this could be made gentler by some unpacking, let me think about it some more (or please make a suggestion if you have one). --JBL (talk) 17:37, 1 August 2025 (UTC)

Ok, I see the intent now behind this phrase! I think we could rewrite it as "The collection of all possible intersections among the hyperplanes in the reflection arrangement..." This makes it clear where those hyperplanes are coming form, and reuses the terminology established in the previous sentence. Gramix13 (talk) 19:21, 1 August 2025 (UTC)

Yes, that's better, thanks -- done verbatim. --JBL (talk) 20:07, 1 August 2025 (UTC)

In the case of a finite real reflection group, this definition differs from the classical one, where S necessarily comes from the reflections whose reflecting hyperplanes form the boundaries of a chamber. What chamber is this referring to? Gramix13 (talk) 00:04, 20 July 2025 (UTC)

Being held accountable for things I wrote in efns, yikes, brutal ;-p When you take all the reflections in a finite real reflection group, their hyperplanes divide space up a bunch of identical pieces (e.g., in the case of a dihedral group coming from an n-gon, you get 2n pizza slices) -- the technical name for these is chambers. Here's my attempt to gloss it (maybe could be better?): --JBL (talk) 17:44, 1 August 2025 (UTC)

I like the example of the dihedral group of the $n$-gon used to explain a chamber. I think that should be included as an example. At that point the efn might get too long (and sorry for brutally critiquing it), so it might be best to combine it with the rest of the paragraph. Gramix13 (talk) 20:00, 1 August 2025 (UTC)

I've rearranged and rewritten, moving the footnote to attach to the discussion of the dihedral group in the next paragraph . Thoughts? --JBL (talk) 19:26, 3 August 2025 (UTC)

Took me a couple minutes to digest what this was saying (I needed to make a visual to see that the description of the slices was correct), but I think this is definitely much better. One change I'd recommended is in the sentence but one could instead choose as S one of the other pairs of reflections, which is not conjugate to the pairs coming from a chamber. I think this should additionally specify that the pair is again one that is the same angle apart bounding a slice (unless I am mistaken as to what pair this should be). Gramix13 (talk) 19:47, 3 August 2025 (UTC)

How about now? --JBL (talk) 17:31, 4 August 2025 (UTC)

That small additional clarity helps there, it should be good now. Gramix13 (talk) 18:49, 4 August 2025 (UTC)

When W is an affine Coxeter group, the associated finite Weyl group... These two terms should be defined for the reader. Gramix13 (talk) 00:07, 20 July 2025 (UTC)

Added gloss; needs citations, though. --JBL (talk) 18:40, 1 August 2025 (UTC)

That's a good description to Weyl groups. One comment I have is we already have used "lattice" for the order structure and now the geometric/group structure, so maybe consider clarifying the type of lattice we mean in the article each time the word is used to avoid confusion if the reader doesn't check the links (no explanation/definition for the lattice, just identification to disambiguate). Gramix13 (talk) 20:04, 1 August 2025 (UTC)

Thanks -- I had intended to do this originally and forgot to come back to it. I've glossed in a way that is not completely precise but which hopefully succeeds in the key task of warning people that this is not the same kind of lattice as discussed elsewhere; totally open to tinkering. --JBL (talk) 19:34, 3 August 2025 (UTC)

That looks good. I was thinking of replacing "that is" with "in this context" to be more precise that the term lattice in that section is different from how the word is used outside of it to mean a special type of ordering. Gramix13 (talk) 19:51, 3 August 2025 (UTC)

I tried a different tweak, moving the technical term into the parenthetical instead of the other way around . --JBL (talk) 21:13, 3 August 2025 (UTC)

Looks fine by me! I don't think we need to change this any further. Gramix13 (talk) 21:18, 3 August 2025 (UTC)

Now added citations for this material . --JBL (talk) 19:46, 3 August 2025 (UTC)

The definition of crystallographic Coxeter group in the footnote doesn't quite seem clear to me, what is the natural geometric representation, and what does it mean to stabilize a lattice? Gramix13 (talk) 00:09, 20 July 2025 (UTC)

So for stabilize a lattice I've now glossed this in the previous paragraph and it means the same thing here. As for natural representation: ugh our Coxeter content is so incomplete that there's nowhere on Wikipedia to point to for this. The answer to the question "what is the natural geometric representation?" is that one can build (in a natural way) for every Coxeter group a space on which the group acts as a reflection group (in an appropriate sense); this is done over a few pages in Bjorner and Brenti (sections 4.1 and 4.2) or Humphreys (sections 5.3 and 6.2). I am not sure this is something I can gloss meaningfully, and expanding on it at length would be undue, in my opinion (it's a minor point). I'll come back and think about it again, but at present my preference order is

leave it somewhat cryptic (people can read the reference) > remove it >> try to explain it in sufficient detail.

--JBL (talk) 18:55, 1 August 2025 (UTC)

Alright, I think with the clarification from earlier, we can probably leave it as is. I think if we mention the $D_{8}$ example again and how it stabilizes a square lattice, we can maybe let the reader know that the square lattice is the group's geometric representation (assuming I understood that right) so they can a solid example to think on. Gramix13 (talk) 20:09, 1 August 2025 (UTC)

How do you feel about it now ? --JBL (talk) 21:12, 3 August 2025 (UTC)

The section titled Connection with the theory of algebraic groups should spend time explaining the terminology used in the section, such as algebraic groups, Borel subgroup, (B, N) pair, Bruhat decomposition, and possibly Double coset. None of these terms would be immediately understandable to an undergraduate reader. Gramix13 (talk) 00:15, 20 July 2025 (UTC)

None of these terms were understandable by me, a professional mathematician, until relatively recently ;). I have made some attempts, please feel free to tell me to do more. I will need to check if the present citations (page-ranges) cover this material. --JBL (talk) 20:29, 3 August 2025 (UTC)

I really like the approach here of introducing algebraic groups and their related terms with the example of the general linear group, and then generalizing in the case of algebraic varieties. Even if an undergraduate might not know what an algebraic variety is, they can keep the picture of the general linear group in their head, while those who are more versed in algebraic geometry can use the algebraic variety description as the more complete way to describe them.

If G is an algebraic group and B is a Borel subgroup for G, then a parabolic subgroup of G is any subgroup that contains B. Just to clarify, are these parabolic subgroups dependent on the choice of $B$, or are we just considering here any subgroup that contains a Borel subgroup to be parabolic? From reading the rest of the section, it doesn't look like the definition is dependent, but if I am misunderstanding this then please let me know.

In terms of the $(B,N)$ pair, the $N$ is left unexplained, I think that should be specified as being essential to understanding the Weyl group of $G$. Moreover, mentioning the special role $N$ plays in this pair would help show why we can form the Weyl group (the intersection $B\cap N$ being normal in $B$). I think also taking the time to mention the generators of the Weyl group and their properties that must hold from the $(B,N)$ pair would help the reader understand how we form the Coxeter system.

Overall, this section is definitely written much better than when I read it the first time, this feels much more tangible to read through and I mostly understand the core idea of what its trying to communicate. Gramix13 (talk) 21:16, 3 August 2025 (UTC)

I think what would be additionally nice would be to show an example Weyl group of the general linear group, using the Borel subgroup of lower triangular matrices. This would help give a concrete example of what such a group would look like to the reader, and it also continues to build up the prior examples of algebraic groups and Borel subgroups. Gramix13 (talk) 21:21, 3 August 2025 (UTC)

Just to clarify, are these parabolic subgroups dependent on the choice of $B$, or are we just considering here any subgroup that contains a Borel subgroup to be parabolic? For each Borel subgroup, the subgroups that contain it are standard parabolics relative to B; different B give you different standard parabolics. But all the different Bs are conjugate to each other (at least if my field is algebraically complete), and so the set of parabolic subgroups (the conjugates of the standard parabolics) is the same regardless of the choice of B. I will revisit this.
The problem with N is that its definition requires yet another definition (of maximal torus) that I would really love to avoid getting into. I will think about it.
Your suggestion to extend this example is excellent, actually I can go as far as giving all the standard parabolics of GL_4 and seeing how they correspond to the standard parabolics of S_4, which already appear in a figure above. --JBL (talk) 23:22, 3 August 2025 (UTC)

Apparently the Digne--Michel reference gives a wrong definition of W in the context of a (B, N)-pair (either that or I've made a mistake in transcribing) and actually the torus is unavoidable; I'll need to double-check this with other sources tomorrow. --JBL (talk) 00:32, 4 August 2025 (UTC)

Well that's unfortunate, hopefully attempting to explain the torus and $N$ doesn't lead to an unnecessary side tangent in the article in that case. Gramix13 (talk) 00:43, 4 August 2025 (UTC)

This section has now blown up in size but I think it's worth it -- in particular, I'm comfortable that this amount of content is in fact WP:DUE. I was writing from memory (I didn't actually make it into the office today) so I will have to go back, make sure I haven't made any egregious errors, and add citations. --JBL (talk) 18:38, 4 August 2025 (UTC)

The example there is great, and I love how the parabolic subgroups are illustrated as well so you can see what entries are allowed to be free in that group. The only other thing I want to suggest would be to mention a generic field that the general linear group is taking entries in just for the sake of being formal and precise, but that's really it. I might also suggest splitting the parabolic subgroups into four lines, with two subgroups per line, just to make it easier to read on mobile devices. Gramix13 (talk) 18:59, 4 August 2025 (UTC)

Done and done. --JBL (talk) 19:59, 4 August 2025 (UTC)

In particular, the group W acts on the complement of the complexification of the arrangement of its reflecting hyperplanes; the generalized braid group of W is the fundamental group of the quotient of this space under the action of W. All of this could be elaborated and expanded into its own paragraph, this clarification would especially be useful for an undergraduate who might not be as familiar with algebraic topology and would be unfamiliar with these terminologies. Gramix13 (talk) 00:21, 20 July 2025 (UTC)

I want to argue that I have carefully hidden all these technicalities away in an efn precisely because they are not meant for a general reader (whatever that means in this article) but only for relative experts, and that I should be allowed to get away with that once or twice in an article of this size. (Unlike the previous point, on algebraic groups, I do not think a major expansion in the exposition here would be due weight -- this whole section is a minor aspect IMO.) --JBL (talk) 18:42, 4 August 2025 (UTC)

Yeah, looking back at this, I agree with you that this side tangent would be undue, the efn would be fine as is. If someone really wanted the details of what that means, then that goes beyond the scope of the article, and they could look into studying algebraic topology for that kind of information. Gramix13 (talk) 19:03, 4 August 2025 (UTC)

The above replies conclude my comments on the understandability of the article. I mainly commented on terms that the article doesn't explain which an undergraduate reader might reasonable not understand coming into the article nor be expected to know. I think if these are all addressed, including in the section titled Connection with the theory of algebraic groups, then I believe criteria 1a will have been met for this article. I hope the reviewer and nominator find these comments helpful, and I am willing to elaborate on them if there's any questions on what I have said. Gramix13 (talk) 00:26, 20 July 2025 (UTC)

@Gramix13: Thanks so much for this detailed review! Kingsif (talk) 01:05, 20 July 2025 (UTC)

Agreed! I have begun working my way through them; having guests over so whatever I don't get to in the next hour will have to wait until Monday probably. --JBL (talk) 18:28, 1 August 2025 (UTC)

Thanks for the work so far! Kingsif (talk) 01:55, 2 August 2025 (UTC)

@Kingsif and Gramix13: With the exception of fixing the sourcing issue I've introduced in the section Parabolic_subgroup_of_a_reflection_group#Connection_with_the_theory_of_algebraic_groups (which I will get to later this week when I have some references in hand), I want to say that I've addressed all the comments above. Please point out if I've missed something. --JBL (talk) 18:46, 4 August 2025 (UTC)

Looks like all of my comments have been addressed. I think I'm really satisfied with how this article has turned out from this review. I definitely think this article well meets criteria 1a. I think its safe for Kingsif to wrap up this review unless they have other concerns about the article, but from the comments here I doubt that would be the case. Gramix13 (talk) 19:15, 4 August 2025 (UTC)

Yes, going through the comments that I follow, the article has been well updated. Clarity is provided and certain subjects have been expanded on. It reads logically from start to finish, while all sections are sufficient as standalone. I will wait on confirmation of sources in this edit from @JayBeeEll: before final pass (meeting crit 2c), but I have no other concerns with regards to the criteria: 1b is met for topic, 2a the article uses a standard ref format, 2b all sources on the matter are high quality, 2d no evidence of CV, 3a and b have been covered in the comments here, and 4 and 5 easily observable. Thanks so much guys. Kingsif (talk) 23:12, 6 August 2025 (UTC)

Excellent, thank you both -- I will report back on the sourcing in that section tomorrow. --JBL (talk) 00:04, 7 August 2025 (UTC)

@Kingsif: I have now fully corrected and referenced everything in the expanded section on algebraic groups. (Full disclosure: the concrete example is referenced in a way that is recognizable as verification only to a rather mathematially sophisticated reader, because the source is written for a graduate and research audience.) --JBL (talk) 00:59, 8 August 2025 (UTC)

That it's verifiable is important, not necessarily that this needs to be accessible and understandable to everyone. We accept non-English sources and I see this as no different in practice. Only the article itself being accessible and understandable to its appropriately broad audience is required. Kingsif (talk) 22:37, 8 August 2025 (UTC)

I don't agree that this is a subject that is introduced at the undergraduate level. I opened two textbooks I owned on abstract algebra, both of which were mainly written for a graduate level, and I couldn't find a solid reference to parabolic subgroups, let alone Coxeter groups which is vital to the definition of parabolic subgroups. Dummit and Foote's Abstract Algebra has only a mere passing reference to them and certainly not one that would be expected of reader to remember and study. I believe JBL's intent was that the topic is introduced to PhD students, specifically to those who might want to study group theory, but the article itself is written one level down of that to be accessible to undergraduates (particularly those who have already done group theory).

While I agree that we shouldn't be removing content just because it might be above the level of an expected reader, that doesn't really mean we shouldn't make some effort to try explaining some of the content and terminology that is higher level that the reader may be unaware of. This would give a stepping stone into the content rather that a cliff that they would have to go around to find a way up. That's why in the comments above I focused on lingo that wasn't elaborated in the article as I felt it implied that the reader would be expected to understand what that meant, and if they have to follow a blue link to an article explaining those terms then the main article might not be succeeding in being understandable. With that said, the rest of this article is for the most part understandable aside from those hiccups I mentioned, and I don't really think those bumps really take away from delivering the key areas of the topic, so I don't think my points should greatly discount this article from meeting criteria 1a.

This is just me throwing my own two cents on the matter and hope it helps the reviewer at least to put this content in some perspective. I haven't performed any full GA review, nor have I gotten an article through GA, so feel free to take all this with a grain of salt if you wish. Gramix13 (talk) 17:16, 20 July 2025 (UTC)

As you describe your understanding of the intent is how I (tried to) explained it, I feel: when I wrote "the academic level of an undergraduate" I phrased it that way and precisely did not say 'undergraduate level' to distinguish between the academic abilities of an undergraduate student (intention) and the curriculum (not). Kingsif (talk) 21:16, 20 July 2025 (UTC)

• 6b: illustrations and relevance to topic
• I have a suggestion to move the image of the dihedral group $D_{8}$ from the section In complex reflection groups to Background: reflection groups since I think it would help establish an early picture to the reader of what a reflection group could look like, and I think dihedral group would be an excellent example for this. Some of the images in Dihedral group could also be useful for this section too if the image I suggested really should stay in that spot, in particular there's some showing the axes of reflection, and another showing reflections causing a rotation. Gramix13 (talk) 22:51, 19 July 2025 (UTC)

• Do you think a gallery would be useful to illustrate different reflection groups? Also noting I have moved this comment to a section on illustration. Kingsif (talk) 22:54, 19 July 2025 (UTC)

  I was going to suggest that those illustrations would better served in the Reflection group article, until I then noticed there are absolutely no images in that article at all! Such a gallery would be much more suited to that article which could really use some illustrations. So no, I don't think a gallery of reflections would be necessary for this article.

  I still think at least one image in the background section for the dihedral group would at the very least support the reader in getting a picture for what these reflection groups should look like and build on the intuition on what a reflection is. It would also help them understand why we want $ss=1$ for some generator $s$ of a coxeter group, because these should be thought of as reflections in the context of a reflection group, and it should be intuitive that doing a reflection twice does not change the orientation of the object, or in other words is the identity action. (take your phone and flip it on an axis, then do that same flip again, is the orientation of your phone now the same as before you did this exercise?) Gramix13 (talk) 23:03, 19 July 2025 (UTC)

Having an image of a Coxeter-Dynkin diagram could also help illustrate parabolic subgroups as the caption could explain how to identify those subgroups from the image. I imagine having such an image in the definition section under Coxeter groups. Gramix13 (talk) 23:22, 19 July 2025 (UTC)

I am noticing that the lattice diagrams have the full expressions of each group but that they are intersection with the inclusion lines of the diagram. Is there a way to move them or change their size in a way that prevents that issue and makes the images more clear to read from? Gramix13 (talk) 00:00, 20 July 2025 (UTC)

I don't understand what this comment/question means. Can you clarify? --JBL (talk) 18:26, 1 August 2025 (UTC)

The image of $D_{8}$ in the article is a good example of what I mean. Notice the text next to each of the middle four dots (the equations involving x and y) have lines going through the text, and that can make it difficult to read. Notice for example how the yellow equation has a line which happens to go directly through the equal sign, making it almost read like $\neq$. This isn't the only image with that issue, other ones like it have issues where those lines will intersection with the equations. If the equations were moved to avoid those lines, or maybe the lines hide underneath the equations to avoid disrupting them, that would address the issue. Hopefully that clarification helps, if that doesn't make sense feel free to ask more clarifying question so I can better explain. Gramix13 (talk) 20:16, 1 August 2025 (UTC)

Ok, I get it. I'll try mocking up a few different versions in the next couple days and we can see what looks best. I do know that I didn't like putting the labels on the same line as the dots (which avoids the problem you note) because it was hard to tell whether labels went with the dot on the left or on the right. --JBL (talk) 23:22, 2 August 2025 (UTC)

Would be nice to have a picture of the Braid group in the titular section to illustrate what happens when we remove the $s^{2}=1$ relation. The image used in that article's lead might be a good candidate. Gramix13 (talk) 00:18, 20 July 2025 (UTC)

Good images are hard and take a lot of work. I don't see a way to draw the first and third images currently in the article to avoid the issue with group names crossing poset edges, at least as long as the picture is going to give the same amount of information. Here are two that are relatively easy, given what I've already done:

The reflection arrangement of D_2×5. One chamber is shaded gray. Any two reflections form a Coxeter generating set, but the pair s, t is not conjugate to the pair r, t.

New version of something already there; better?

I can imagine a good image that showed all 8 elements of D_2×4 via their action on a labeled square, and how they corresponded to the Coxeter structure, but it will take me several hours to create (that I don't have available at this moment). Braid images are very hard, but for the "usual" braid group (of the symmetric group) there are good ones in Braid group that I could steal. I could make a version of the first image in which the vertices of the Boolean lattice are labeled by the Coxeter--Dynkin diagrams of the parabolic subgroups, although I'm not entirely convinced how much that would add; maybe the information currently in that first image could be split over two images (one showing the groups explicitly but not in a lattice, leaving a simpler lattice without the edge-crossing issue)? Also will require a fair amount of time, though. --JBL (talk) 23:22, 3 August 2025 (UTC)

I think all of this is good, although if a new image will take time to produce, I wouldn't worry about having that image completed in time for the sake of GA since I think the article's illustrations already suffice for the criteria (but of course that is the reviewer's job to make that determination, this is my own two cents as a bystander). Gramix13 (talk) 00:39, 4 August 2025 (UTC)

I put both of these images into the article (well in one case just a replacement) and added a new image of the dihedral group in the background section. --JBL (talk) 18:47, 4 August 2025 (UTC)

New image there is perfect for describing the dihedral group on a square, no further comments from me on images. Gramix13 (talk) 19:05, 4 August 2025 (UTC)

Overall

• Spot check pass. Kingsif (talk) 22:05, 19 July 2025 (UTC)
• The citations are all pretty consistent, but some lack page numbers. Kingsif (talk) 22:05, 19 July 2025 (UTC)