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) Any editor who has not nominated or contributed significantly to 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

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)[reply]

I have added something about the name based on the MO thread. --JBL (talk) 22:00, 16 February 2024 (UTC)[reply]

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)[reply]

@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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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 ${\displaystyle (W,S)}$.

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 ${\displaystyle (W,S)}$, with a specific generating set S tacitly understood. Some caution is necessary in such cases, since the isomorphism type of ${\displaystyle (W,S)}$ is not determined by the group W alone.

Our Coxeter groups article gives a concrete example: the Coxeter groups of type ${\displaystyle B_{3}}$ and ${\displaystyle A_{1}\times A_{3}}$ 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 ${\displaystyle S_{n}^{B}}$. When n = 3, its group structure is insufficient to identify its standard parabolic subgroups: interpreting the group as the Coxeter system ${\displaystyle B_{3}}$ yields a Boolean lattice of 2^3 = 8 parabolic subgroups, whereas interpreting it as the Coxeter system ${\displaystyle A_{1}\times A_{3}}$ 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 ${\displaystyle \{1,\ldots ,n\}}$) has maximal standard parabolic subgroups the stabilizers of ${\displaystyle \{i+1,\ldots ,n\}}$ for ${\displaystyle i\in \{1,\ldots ,n\}}$ (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 ${\displaystyle \operatorname {Stab} _{G}(x)}$ of a point x, but does not define what the stabilizer subgroup ${\displaystyle \operatorname {Stab} _{G}(X)}$ 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 ${\displaystyle \operatorname {Stab} _{S_{n}^{B}}(\{i+1,\ldots ,n\})}$ means to clarify which definition is being used here. Preimage (talk) 13:14, 6 May 2025 (UTC)[reply]

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 ${\displaystyle g(A)=A}$ and ${\displaystyle g(A)\subseteq A}$, 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)[reply]

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

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

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

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