Wikipedia:Featured article candidates/Affine symmetric group/archive1

Affine symmetric group

 * Nominator(s): JBL (talk) 17:20, 30 June 2023 (UTC)

This article is about a mathematical object that is of interest to pure mathematicians in a wide array of areas. I believe this article presents a comprehensive account of its subject, including its multiple definitions (and why they are equivalent), its many interesting properties and substructures, and its substantial connections to other mathematical objects (especially the "usual" finite symmetric group of permutations, which appears in nearly every corner of mathematics). While the affine symmetric group is not usually encountered outside the context of research mathematics (say, by PhD students or professional researchers), I believe the article is written so that significant portions of it can be appreciated by readers with a more modest mathematical background, and nearly all of it appreciated by an undergraduate who has taken a first course in group theory. This is my first FA nomination, but I received extremely helpful guidance from before submission. JBL (talk) 17:20, 30 June 2023 (UTC)

First-time nomination

 * Hi, and welcome to FAC. Just noting that as a first time nominator at FAC, this article will need to pass a source to text integrity spot check and a review for over-close paraphrasing to be considered for promotion. Good luck with the nomination. Gog the Mild (talk) 18:00, 30 June 2023 (UTC)
 * Just commenting here to say that I was the one who approved the article for GA status and didn't notice any source-to-text integrity issues at the time. Further spot-checking should still be done; I'm pretty confident that the material is all in the books that it's claimed to be in, but page-number errors can catch the best of us. One quick note about the prose: in § Algebraic definition, perhaps it would flow more nicely to say what the vertices and edges of a Coxeter–Dynkin diagram mean before saying what the diagrams of the affine symmetric groups are. XOR&#39;easter (talk) 21:39, 2 July 2023 (UTC)
 * Yeah you're absolutely right, the flow is funny there. That is a good suggestion, I will adjust.  --JBL (talk) 23:13, 2 July 2023 (UTC)
 * I did a thing, hopefully it's smoother now. --JBL (talk) 17:44, 3 July 2023 (UTC)
 * Thanks . If you have done the donkey work, do feel free to finish it off here. And/or do a standard source review. :-) Gog the Mild (talk) 19:01, 4 July 2023 (UTC)

Coordinator note
This has been open for three weeks and has yet to pick up a support. Unless it attracts considerable movement towards a consensus to promote over the next three or four days I am afraid that it is liable to be archived. Gog the Mild (talk) 12:16, 21 July 2023 (UTC)

Some random comments
It's asking a lot and I won't insist on it, but is it possible to footnote some of the jargon used in the article? Wikipedia mathematical articles are often incomprehensible to outsiders and this one doesn't seem to be an exception. What I have to insist on is that we avoid one or two line long paragraphs and unsourced sentences, of which there are some. As well as "we" language. Jo-Jo Eumerus (talk) 09:02, 24 July 2023 (UTC)


 * Hi, thanks for your comment. Reading mathematics is hard for basically everyone at all levels, including professional mathematicians, because of the level of abstraction and the density of information inherent in mathematical notation.  Some of this is avoidable and some is not.  Could you give a small number of examples of jargon that you think could be clarified reasonably by a footnote?  (I.e., illustrative rather than comprehensive.)  It would be helpful if you could provided detailed comments on the introductory section (preceding Definitions), for example.
 * I believe I have fixed the unique instances of the first-person plural -- please correct me if I am wrong. (As Iry-Hor mentions, this is the standard style in mathematical writing, so even when I'm being vigilant a few sneak through.)
 * What sentences do you believe are unsourced?
 * I quickly glanced through again, and the only one- or two-sentence paragraphs I noticed are introductory paragraphs at the beginnings of some multi-part sections, that summarize at a high level the contents of the section that follows. Personally I find such brief instances of guiding text extremely helpful when trying to understand writing on any technical topic; if you object to this, it would be helpful if you could express what countervailing principle you feel applies more strongly.  (Or maybe your comment is not about those paragraphs, but about some others I overlooked?)
 * All the best, JBL (talk) 17:47, 24 July 2023 (UTC)
 * I've given it another read. The only thing I've found that ought to be cited is in the "Geometric definition" subsection: However for higher dimensions, the alcoves are not regular simplices. Section 4.3 of Humphreys talks about alcoves that are not equilaterial triangles, but it doesn't say anything specific about higher dimensions. XOR&#39;easter (talk) 21:13, 24 July 2023 (UTC)
 * Thanks, I will try to run it down. --JBL (talk) 00:19, 26 July 2023 (UTC)
 * At the moment I haven't found a good source explicitly saying this (it's a standard thing to say in a classroom setting, maybe it doesn't get written in books or papers), so I've removed it. --JBL (talk) 17:20, 30 July 2023 (UTC)
 * There are several paragraphs with no source at the end - they should either get one or be merged with the following paragraph. I also don't think that "we" language is accepted style on Wikipedia. Jo-Jo Eumerus (talk) 07:17, 25 July 2023 (UTC)
 * How close to the end should we be looking? "History" is one paragraph, cited; "Relationship to other mathematical objects" has at least one footnote per paragraph, apart from the line at the top that just summarizes the section to follow. Am I overlooking something? (That's always possible.) "We" language is generally more textbook-like than Wikipedia house style prefers, but this edit seems to have gotten the last of it. I did a find-in-page just now for we as a whole word and got no results. XOR&#39;easter (talk) 18:27, 25 July 2023 (UTC)
 * Last sentence before the paragraph break. Jo-Jo Eumerus (talk) 19:15, 25 July 2023 (UTC)
 * In which paragraph? XOR&#39;easter (talk) 19:58, 25 July 2023 (UTC)
 * The situation when is shown in the figure; in this case, the root lattice is a triangular lattice, with reflecting lines dividing it into equilateral triangle alcoves. However for higher dimensions, the alcoves are not regular simplices. and For example, a portion of the matrix for the affine permutation is shown in the figure for example. Jo-Jo Eumerus (talk) 16:42, 26 July 2023 (UTC)
 * The former is cited to a book (the part not explicitly spelled out in that section of that book has been removed), and the latter is an example of a technique explained in the cited source, Chmutov, Pylyavskyy & Yudovina (2018). I'm still not seeing the problem. Anyone who has gotten so far into a math degree that they've taken a group theory course will be capable of looking a sentence backwards for a footnote. XOR&#39;easter (talk) 21:51, 31 July 2023 (UTC)
 * It is not however accepted Wikipedia style to make readers go backward in the paragraph to find the citations. Maybe that's accepted style in mathematics textbooks, but Wikipedia is notionally supposed to work for a general audience as well. Jo-Jo Eumerus (talk) 09:04, 1 August 2023 (UTC)
 * A "general audience" is not going to be reading multiple sections down into affine symmetric group. They're just not. Most of the are going to nope out exactly nine words into the article, when it says "mathematical". WP:CITEFOOT says The citation should be added close to the material it supports; the Featured article criteria do not go into detail at all; the recently beefed-up Good article criteria say that content must be cited no later than the end of the paragraph but do not rule out earlier. Maybe the footnote placement could be jiggled or a citation repeated somewhere, but I see little if any actual benefit and no reason grounded in policies and guidelines to draw a hard line. XOR&#39;easter (talk) 17:30, 1 August 2023 (UTC)
 * Hi Jo-Jo Eumerus,
 * Thanks for your additional comments, and sorry for the delay in responding to them. With respect to the three sentences you quoted above (in this comment): the one that begins "However" has been removed after I failed on first attempt to find a source for it.  For the third one ("For example") it is not referenceable in principle, because it is a statement about a figure that I created for the article, which does not exist anywhere else.  However, I would like to appeal in this instance to WP:CALC: I think the assertion in that sentence (that the figure has the properties it does, and how those relate to the preceding cited text) is a routine calculation (bearing in mind that Mathematical literacy may be necessary to follow a "routine" calculation).  The other one is a bit more subtle and I will get back to you on it later.  --JBL (talk) 00:59, 2 August 2023 (UTC)
 * I have added two sources on the sentence about the triangular lattice (as well as adding some more information). This was difficult to do because in the research literature, authors generally treat this as an instance of WP:CALC, expecting readers to be able to fill in the details themselves.  (See, for example, the treatment of Figures 6 and 9 in this paper: a general definition is stated, and then this particular case is illustrated, in the expectation that the reader will be able to verify from the general definition the salient features of the specific case.)  I called it "subtle" because I think from the point of the Wikipedia audience the application of WP:CALC in this case is dubious. Between the two sources, however, I think it is now adequately sourced.  --JBL (talk) 18:36, 4 August 2023 (UTC)

Support by Iry-Hor
, I reviewed this article some time ago and had not noticed it being here owing to being busy outside of Wikipedia. I strongly support this article: it is well written and comprehensive. Now all experienced editors have noticed that there are very few FA maths articles and technicality is one of the reasons: I do not think it is possible to really make an FA quality article on such a subject without being very technical, and I think no amount of hand-waving will transform this matter into universally understandable concepts without compromising the exactness of what is being said. Contrary to geography or history, which are much easier to convey to non-specialists, I really think that this is not the case for advanced maths. Also, the "we" style is typical of scientific literature. I emphasize that Wikipedia has a real maths problem: not enough article and too few of FA standard yet this one is clearly of FA quality. The subject is difficult and its readers will be mathematicians and students of maths, who need a coherent source synthesizing the subject and this is it.Iry-Hor (talk) 16:19, 24 July 2023 (UTC)
 * Thanks Iry-Hor. At the moment the article is in dire need of reviewers. Perhaps might be enticed into writing one? Or perhaps you or JBL might place a polite neutrally phrased request on the talk pages of a few of the more frequent FAC reviewers, or on the talk pages of relevant Wikiprojects, or of editors you know are interested in the topic of the nomination. Gog the Mild (talk) 20:56, 24 July 2023 (UTC)
 * Since I was pinged ...
 * Between administrative work and the content I'm working on, I have a lot to do, but I will try to take a look. Briefly swinging through it it looks better than it did during the GA process (which, no fault of the nominator, I do not have fond memories of). Daniel Case (talk) 05:05, 26 July 2023 (UTC)

Comments by Shapeyness
Hi JBL, I'm not particularly knowledgeable on pure mathematics so I probably won't be able to give comprehensive comments on the whole article, but here are some suggestions/things to think about. Shapeyness (talk) 16:04, 25 July 2023 (UTC)
 * Hi, thanks very much for your comments! I will respond inline below (first batch now, more to come).  --JBL (talk) 21:16, 25 July 2023 (UTC)


 * On first reading, I found quite hard to understand. Perhaps it is better to introduce the concept of of a finite symmetric group, and then (in a separate sentence even) explain that affine symmetric groups all extend from a finite group to the infinite case.
 * Okay I think that this plus the next bullet point were really caused by this sentence being in the wrong place. I've moved it to the following paragraph and split it into two parts there.  I hope that improved flow makes both paragraphs easier to follow (and please let me know if not!).  --JBL (talk) 00:24, 26 July 2023 (UTC)
 * This is far better yes, and makes even some of the unchanged sentences a lot clearer! Shapeyness (talk) 13:22, 26 July 2023 (UTC)
 * what geometric description? Is this talking about the idea that they are ?
 * Yes, that was the intention. --JBL (talk) 21:16, 25 July 2023 (UTC)
 * A small signpost-y sentence might also be useful here, e.g..
 * In general, the lead is quite short, I think there is room here for some more explanations and extended details (assuming there is content from the body of the article to draw from for this). For reference, MOS:LEADLENGTH suggests a lead of three paragraphs of about 300 words, and up to four paragraphs if needed. This is not necessarily needed but might help.
 * Also, keep in mind that the lead should be the most accessible part of the whole article to give a general idea to as large an audience as the subject allows. Not suggesting any changes on this but something to think about if you do decide to expand it or make any changes.
 * (Responding to all the points above.) Yes, you're right, I have a bit of room to expand the intro.  I need to think a bit about how best to do it, but I definitely can unpack finite symmetric groups and warn that there are multiple characterizations coming, for example.  --JBL (talk) 21:16, 25 July 2023 (UTC)
 * I find this slightly hard to understand - I think the idea is that the generators together with the relations imply all other group elements? Or the relations imply all other equations which hold for all group members? Sorry, bear with me if this is just my mathematical ignorance coming through.
 * Oof yes this sentence is terrible. To unpack it here: the property of being generators means that every other group element can be written in terms of them.  The relations tell you that sometimes different expressions in terms of the generators are equal (just like 2 + 5 and 4 + 3 are different expressions for the same quantity).  The confusing part is the assertion that every relation satisfied by elements of the group is a consequence of the given relations.  (This package of three things -- generators, relations, and the fact that every relation is a consequence of the given ones -- is the meaning of the phrase "definition of a group by generators and relations".)  Let me think about it.  --JBL (talk) 00:32, 26 July 2023 (UTC)
 * Would something like work? Just a suggestion and probably not the most elegant way of putting it. Shapeyness (talk) 13:29, 26 July 2023 (UTC)
 * I did a thing, pushing the technicality into a footnote (where it can be expanded on without burdening the text). --JBL (talk) 17:56, 30 July 2023 (UTC)
 * Why is this in brackets?
 * It struck me as a minor point -- in particular, the word "braid" is only used in one other section of the article. I would be happy to either remove the brackets or make it a footnote, if you think one of those would be better.  --JBL (talk) 21:16, 25 July 2023 (UTC)
 * No that's OK, I just wondered, partly also about whether there was a broader relevance to this point. If it is only a minor thing though then no need to change anything. Shapeyness (talk) 13:38, 26 July 2023 (UTC)
 * Ok, thanks. I would say the broader relevance is precisely that these are the relations that are relevant when one considers braid groups (the topic of that other section of the article).  --JBL (talk) 17:56, 30 July 2023 (UTC)
 * The history section is quite short, does it cover all the major points from reliable sources?
 * As it happens, I mentioned the brevity of the history section during the GA review, and it seems there's just not the material to expand it. XOR&#39;easter (talk) 18:21, 25 July 2023 (UTC)
 * Yeah, sadly there seems to be little discussion of the history of this object particularly in reliable sources :(. --JBL (talk) 21:16, 25 July 2023 (UTC)
 * Ok, thanks, just wanted to make sure. Shapeyness (talk) 22:18, 25 July 2023 (UTC)
 * Just a thought: the "History" section currently says that the combinatorial description was expanded upon by Shi (1986). Perhaps a line could be added about what exactly Shi contributed beyond what was already known at that point. XOR&#39;easter (talk) 21:46, 4 August 2023 (UTC)
 * God we mathematicians are such awful historians :). Bjorner and Brenti say nothing more than that one fragment of an idea, and most papers that do any history at all seem to follow their lead.  After spending a couple hours on this beautiful Saturday afternoon looking over the literature again, I am happy to report that Green offers a substantive assertion on this topic, and so I was able to expand the section (very slightly).  --JBL (talk) 21:15, 5 August 2023 (UTC)
 * No comments on the technical details, but the rest of the article seems well written even if quite a bit of it was too technical for me to fully follow. In general the article is quite technical, which is ok, but think of what level this is generally studied at and try to write a level below that (WP:ONEDOWN). E.g. if this is only delved into at PhD level, then write for an audience of masters students, if it is taught at masters level then write for undergraduate maths students, etc. I don't know what level affine symmetric groups are taught at or what level this wikipedia article could be understood by, but hopefully the article already satisfies this.
 * I tried to address this in my nomination statement. This object does not appear in a typical undergraduate mathematics curriculum; it is usually met only by PhD students or research mathematicians.  I have worked quite hard to make the article accessible to someone approaching the subject with a more limited background (say, an undergraduate who has taken a course in group theory and a course in combinatorics).  Ultimately there are limits on how low-level one can go while staying true to the source material, especially for some later sections (like the one on affine Lie algebras).  --JBL (talk) 21:16, 25 July 2023 (UTC)
 * Ok, that's what I suspected, thanks. Shapeyness (talk) 22:13, 25 July 2023 (UTC)
 * Also, yep, should have read through the nomination statement more thoroughly, sorry about that! Shapeyness (talk) 13:41, 26 July 2023 (UTC)
 * Oh no need to apologize, I'm sure! --JBL (talk) 17:56, 30 July 2023 (UTC)
 * I will attempt another read through soon and give any more comments if I have them.

Ok, few more comments below. Shapeyness (talk) 15:32, 26 July 2023 (UTC)
 * Geometric definition: I found this to be relatively accessible. However, I was a bit confused about what the connection/relevance of the root lattice has to the geometric definition.
 * Yes I see your point. Really the root lattice isn't part of the definition (at least as the definition is presented in our article).  Maybe it shouldn't appear until later -- I'll have to think about it.  --JBL (talk) 17:56, 30 July 2023 (UTC)
 * I also wonder whether the example of n=3 could be expanded to walk through each step of the definition a bit more, I found it useful as a non-mathematician thinking through the definition like this.
 * I spent a lot of time over the last few days looking for the best discussions of the two-dimensional case that I could in the literature, because I agree with you that this would be a better article if the details of that case were worked out explicitly. My experience with that search has been disappointing: after considerable effort, I was able to find sources that explicitly detail the claim that  queried above, but I just can't find authors who go into the necessary detail on this particular case.  As I mentioned above (the comment with the same time-stamp as this one), it just seems like authors of the relevant graduate texts / research literature treat these details as a sort of WP:CALC situation, where readers can work it out for themselves.  I wish I could do better, but unless someone finds a source I've missed (to be clear: possible!), I don't see how to do this without going beyond what is permitted by a strict reading of WP:OR :(.  --JBL (talk) 18:36, 4 August 2023 (UTC)
 * Yes, it's a frustrating situation. XOR&#39;easter (talk) 21:42, 4 August 2023 (UTC)
 * I do think there is an argument for WP:CALC and even WP:IAR here as this clearly wouldn't be unverifiable, but taking a cautious approach for FA also makes sense. Shapeyness (talk) 20:01, 5 August 2023 (UTC)
 * Once this FA is over, if I'm not completely sick of the subject by then :), I might write something up on the article talk-page, just to have a record of it somewhere -- if I do, I'll ping you. Thanks again for all your comments.  --JBL (talk) 21:17, 5 August 2023 (UTC)
 * Connection between the geometric and combinatorial definitions: Perhaps this can also be signposted at the top of the section following, e.g. something like . This just helps assure the reader this subsection doesn't belong to the previous definitions section.
 * This is a nice suggestion, thanks -- I've added a sentence there. --JBL (talk) 17:56, 30 July 2023 (UTC)
 * I think coset and Bruhat order should be linked, representation theory should probably also be linked at first occurrence in the body, also Cartan subalgebra from slightly later on
 * Good comments. I've linked coset, Bruhat order, and Cartan subalgebra. Representation theory is already linked in the lede; I'm not sure whether it needs to be linked at its first appearance in the body (no strong feelings either way). XOR&#39;easter (talk) 17:02, 28 July 2023 (UTC)
 * Thanks both. I've linked rep thry in the first use in the body.  --JBL (talk) 17:20, 30 July 2023 (UTC)


 * Hi JBL. Have you addressed all of Shapeyness's comments? If so, could you ping them to let them know? Thanks. Gog the Mild (talk) 11:22, 1 August 2023 (UTC)
 * Hi, still mulling one over. I will try to get to it in the near future, and will ping Shapeyness when I do -- thanks. --JBL (talk) 22:58, 1 August 2023 (UTC)
 * I think I've responded to all of your comments now. --JBL (talk) 18:36, 4 August 2023 (UTC)
 * Support as the article is now probably as accessible as it ever could be, and I could follow a lot more as a non-mathematician than I would expect given the subject matter. I think the lead is a lot better now and does a good overview without getting bogged down too much into details. I'm not an expert in this at all, but I'm happy from the WikiJournal of Science peer reviews that this is a comprehensive treatment of the subject, and it is well-written throughout. The article is also well illustrated with useful diagrams wherever possible. I think this article definitely deserves to be an FA. Shapeyness (talk) 20:01, 5 August 2023 (UTC)

Comments by XOR'easter
I'm close to supporting this. XOR&#39;easter (talk) 18:03, 1 August 2023 (UTC)
 * Thanks for your comments; I've responded to them below. --JBL (talk) 23:18, 1 August 2023 (UTC)


 * A minor phrasing issue: Coxeter groups have a number of special properties not shared by all groups. These include that they have decidable word problem... I don't know if that's standard phrasing for the area, but it looks a bit like a word is missing.
 * I went back and forth on whether to include an article, and it sounds awkward both ways to me. I've rearranged to "that their word problem is decidable".  --JBL (talk) 23:18, 1 August 2023 (UTC)
 * Core partition is a redlink, but "bounded partition" is not a link at all. Is one of these topics really more article-worthy than the other? I'm fine with redlinks if there's a good case that they ought to be articles; the treatment here just seems a little asymmetrical.
 * Yeah I dunno -- maybe some day I'll write Core partition, for now I've unlinked it. --JBL (talk) 23:18, 1 August 2023 (UTC)
 * Singular versus plural: the article starts with "the affine symmetric groups are" and seems mostly consistent with breaking out the definite article only when talking about a specific group. But it's not wholly consistent. Then the affine symmetric group can be realized geometrically, for example, might be better with a $$\tilde{S}_n$$ inserted to be explicit. Other aspects of the affine symmetric group might be better as Other aspects of an affine symmetric group. Perhaps The affine symmetric group is closely related to a variety of other mathematical objects ought to be The affine symmetric groups are... to parallel how the article begins. I know that saying "the so-and-so group" when there is an infinite family of so-and-so groups is not uncommon, but here it seemed a little more confusing than usual. This isn't a make-or-break issue, but maybe there's a way to improve it with a few small tweaks.
 * As with the previous point, Hyperoctahedral group is used in the singular but is meant in the plural (the statements apply to hypercubes of any dimension).
 * This one is so hard for me because (as you observe) I say things like "the best group is the symmetric group" all the time and everyone understands what I mean even though it's formally nonsense :). The fix for hyperoctahedral group is easy by introduction of notation (which unfortunately is not standard the same way $$S_n$$ is, but we'll manage); I think I've addressed all the instances you mentioned where it applies to the affine symmetric group, either via notation or pluralization.  --JBL (talk) 23:18, 1 August 2023 (UTC)
 * Good edits. Thanks. XOR&#39;easter (talk) 17:53, 2 August 2023 (UTC)
 * Footnotes 13 and 60 are to whole books, Shi (1986) and Lehrer & Taylor (2009) respectively. Could these be made more specific?
 * Oops! I have improved precision on L&T; for Shi, I don't own a copy and Google Books is not cooperating in my efforts to find the best spot, but it's addressed in plenty of sources so I just swapped it out for something else.  --JBL (talk) 23:10, 5 August 2023 (UTC)
 * Support Concerns addressed; not much else to say. XOR&#39;easter (talk) 23:15, 5 August 2023 (UTC)

Source review
OK, I see I've already commented but there was a request for a source review, so I'll put one here on this version. I think the template at the top is somewhat misleading - the linked article was derived from our article, not the other way around. Source formatting and information seems consistent except for the lack of a source link at #33, with all necessary information there but sometimes we are citing page numbers and other times entire chapters. Spot-check but really needs replication from someone who actually knows this kind of mathematics: I can confirm that there is no plagiarism nor close paraphrasing from the sources used, though. Jo-Jo Eumerus (talk) 15:47, 8 August 2023 (UTC)
 * 3: I am not seeing "braid relations"?
 * It seems Bjorner and Brenti don't introduce this terminology until p. 75, and they use a slightly different phrase ("braid-move"). I will fix it. --JBL (talk) 21:23, 9 August 2023 (UTC)
 * I've added a separate reference to support the parenthetical. It's currently commented out because it will change the numbering of all other references, making it impossible to track these comments, but if anyone cares to check it's to p355 of the Stembridge reference already in the article. --JBL (talk) 17:37, 14 August 2023 (UTC) This has changed the reference numbers; consequently I have adjusted all the footnote numbers referred to in your comments below.  --JBL (talk) 19:26, 16 August 2023 (UTC)
 * 4: What piece of information comes from this source?
 * Originally that footnote was attached to the statement "For $$n \geq 3$$, its Coxeter–Dynkin diagram is the $n$-cycle, while for $$n = 2$$ it consists of two nodes joined by an edge labeled $$\infty$$" but it seems clear I have the wrong page-number. I will fix it. --JBL (talk) 21:23, 9 August 2023 (UTC)  I have completely redone the referencing of this paragraph (whose text begins "These relations can be rewritten in the special form ..."), it should probably be checked again.  --JBL (talk) 19:26, 16 August 2023 (UTC)
 * 6 7: This probably need a finer pagination.
 * Oh dear I've now made the tracking of different versions hopeless :(. I agree that finer pagination would be better: in this version the various copies of this reference have been replaced with the more precise references 7, 8, 18, 26.  --JBL (talk) 17:24, 22 August 2023 (UTC)
 * 19 20: What chapter does this refer to? "Affine Weyl group"? A mathematician probably needs to check it.
 * Specifically the first sub-section "The affine Weyl group" (pages 118–119). --JBL (talk) 17:55, 14 August 2023 (UTC)
 * 20 21: What chapter does this refer to? "Construction of root systems"? That does not seem to source anything.
 * Yes, "Construction of root systems". The sentence "For Δ take [equation]" gives (explicitly, but in a different notation) the fact that the generators are $$\{(1,-1,0,\ldots ,0),(0,1,-1,\ldots ,0),\ldots ,(0,\ldots ,0,1,-1)\}$$.  (The specific translation is that $$e_i$$ means the vector that has a single 1 in position i and all other entries 0, so $$e_i - e_{i + 1}$$ is a vector that has a 1 in position i and a −1 in position i+1 and all other entries 0.) That this means that the set is an abelian group and these vectors generate it freely is encoded in the use of the word "lattice" and the meaning of the symbol Δ (i.e., it is not explicit in that section).  --JBL (talk) 17:55, 14 August 2023 (UTC)
 * 21 22: Can't access this source.
 * I don't understand why OAbot decided to mark this as free; I've removed the false tag and added an arXiv link, which you should be able to access -- the section-numbering is the same as in the published version. --JBL (talk) 21:23, 9 August 2023 (UTC)
 * 24 25: Maybe, needs a mathematician to check.
 * 28 29: I don't see the equation there?
 * This is the same as the equation numbered (8.30), up to notational differences between our article and the source. --JBL (talk) 21:23, 9 August 2023 (UTC)
 * 31 32: Maybe, needs a mathematician to check. Don't think we need two consecutive cites to the same thing in the same paragraph. Where is "parabolic subgroup" and "coset" defined - is it background knowledge?
 * I have removed the repeated footnote. --JBL (talk) 21:27, 9 August 2023 (UTC)
 * About the question "where is ... defined?" do you mean in the WP article or in the reference (Bjorner & Brenti) itself? In the reference BB, "coset" is taken as background knowledge, and I followed their lead (just wiki-linking coset).  I've added a separate reference to an earlier spot in BB for the definition of "parabolic subgroup".  --JBL (talk) 19:26, 16 August 2023 (UTC)
 * 32 33: Maybe not ("left"?), needs a mathematician to check.
 * Specifically this is Definition 3.3. --JBL (talk) 19:26, 16 August 2023 (UTC)
 * 33 34: Maybe, needs a mathematician to check.
 * 35 36: The equation for exp ˜S (x; q) is somewhat different from the source.
 * To see that they are the same, one has to substitute into several condensed pieces of notation that are on page 2 of the reference, but there is no substantive difference. --JBL (talk) 21:23, 9 August 2023 (UTC)
 * 36 37: Can't access this source.
 * Same article as footnote 21 22, see response above. --JBL (talk) 21:23, 9 August 2023 (UTC)
 * 46 48: I am not seeing the definition of "tabloid" here.
 * The referenced section (on journal page 67 / PDF page 7) is titled "Tabloids and AMBC" and begins "Given a partition $$\lambda \vdash n$$, a tabloid of shape λ is ..." --JBL (talk) 21:23, 9 August 2023 (UTC)
 * 53 55: Is the equation supposed to be B2 from the source?
 * It's Theorem B1 (b is the number of balls, (a_1 + ... + a_n)/n is the average of the juggling sequence (a_1, ..., a_n) ). --JBL (talk) 19:50, 13 August 2023 (UTC)
 * 59 61: This probably need a finer pagination.
 * 60 62: This probably need a finer pagination.
 * Really? The cited section is only 1.5 pages long.  --JBL (talk) 20:06, 16 August 2023 (UTC)
 * 62 64: Maybe, needs a mathematician to check.
 * 67 69: This probably need a finer pagination.
 * Done (as footnotes 69, 70, 73, 75, 76 in this version). --JBL (talk) 20:06, 16 August 2023 (UTC)
 * 71 77: This probably need a finer pagination.
 * Done (as references 82, 83, 85 in this version). --JBL (talk) 17:59, 22 August 2023 (UTC)
 * Will look at this list soon, but just noting I think the template at the top is somewhat misleading - the linked article was derived from our article, not the other way around. is not right (at least, if I understand to what it refers): the article was drafted in my WP user space, published at WJS, and then imported to WP article space. --JBL (talk) 17:44, 8 August 2023 (UTC)
 * Hi, have you addressed all of Jo-Jo's comments? If so, could you ping them? Thanks. Gog the Mild (talk) 17:13, 16 August 2023 (UTC)
 * To clarify, without a subject-matter expert signing off on the spot-check I am not willing to pass this; too many things seem to require subject-matter knowledge to check. Jo-Jo Eumerus (talk) 19:16, 16 August 2023 (UTC)
 * I've responded to all comments not of the form "This probably need a finer pagination" or "Maybe, needs a mathematician to check", if you'd like to have a look. (I understand that it is necessary to get someone else to look at the last group.) (Actually I have responded to two of the pagination comments now.) --JBL (talk) 20:09, 16 August 2023 (UTC)
 * Brirush's comments below correspond to this version --JBL
 * Footnote 20 has the necessary material and supports the article and is reliable. It represents the material very differently than the article but is mathematically equivalent (i.e. mentions semidirect products instead of projection).
 * Footnote 25 is reliable, and indicates that the space being acted upon is a line, acted on by reflections. The specific details about certain combinations resulting in translation by -2 or 2 is not present in this footnote; however, it may possibly be available in footnote 24, as that refers to a standard abstract algebra textbook and this specific calculation is in every abstract algebra textbook I've seen, but I can't access it to verify. Should be fine.
 * Footnote 32 seems to be represented in a directly straightforward way from a reliable source. Coset is background knowledge; it is very unlikely people would understand anything in this whole article without knowing what a coset is. Parabolic subgroup has been changed to a different footnote so that's fine.
 * Footnote 33: This is a direct translation of the mathematical symbols into english (with w^-1 being referred to as 'inverse'). I accessed on arxiv but seems it was published, with both arxiv and published sources being listed, which is good for access. Seems good.
 * Footnote 34. Can't access this.
 * Footnote 64. Can't access right now but I can try these two again later. Brirush (talk) 16:59, 21 August 2023 (UTC)
 * I found footnote 34 and spent a long time looking over it but just couldn't see how p. 91 applied in this situation. I did some more searching and found a different paper by the author that states this much more clearly (Shi, Jian-yi. "Alcoves corresponding to an affine Weyl group." Journal of the London Mathematical Society 2.1 (1987): 42-55.) page 55 specifically, which is available here: https://math.ecnu.edu.cn/~jyshi/myart/1987JLMalc.pdf. This material is likely in the Shi book referenced in footnote 34 but not at page 91 as indicated. So I recommend switching the source or changing the page number. JBL may have a better understanding here, but the source I listed above explicitly that the length of sw is less than the length of w exactly when certain hyperplanes separate A_0 and A_w.
 * Footnote 64 took a bit as it uses some background knowledge from Lie groups that seems to be general knowledge in the area but wasn't to me (for instance, the references mentions Weyl groups while the text mentions Coxeter groups, which are a superset of Weyl gorups). The version I found online of this reference was missing two pages; however, in what remained, it did clearly state that the Weyl group was generated by reflections of the roots. The source is certainly reliable. I believe that someone who questioned this statement would be able to verify it to their own satisfaction using the source provided.Brirush (talk) 20:18, 21 August 2023 (UTC)
 * Thanks very much for your comments. I don't have access to the 1986 book at the moment, so I can't immediately check what I intended by footnote 34.  I greatly appreciate your going to the effort of finding a better source, and I agree with you that the 1987 paper is extremely clear on this point, so I will replace it in a moment.  --JBL (talk) 17:13, 22 August 2023 (UTC)


 * Hi Brirush, many thanks for looking through this. Would I be correct in saying, in broad terms, that you found no significant issues in the way that the sources had been represented in the article, in so far as you looked at them? Gog the Mild (talk) 12:45, 22 August 2023 (UTC)
 * Yes, I believe the sources are represented accurately in the text and are reliable, and seem to have been carefully chosen. I found no significant issues. Brirush (talk) 19:58, 22 August 2023 (UTC)

Gog the Mild (talk) 21:43, 22 August 2023 (UTC)