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Affine symmetric group

Featured article candidates/Affine symmetric group/archive1

Nominator(s): JBL (talk) 17:20, 30 June 2023 (UTC)[reply]

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 Iry-Hor before submission. JBL (talk) 17:20, 30 June 2023 (UTC)[reply]

First-time nomination

Hi JayBeeEll, 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)[reply]
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'easter (talk) 21:39, 2 July 2023 (UTC)[reply]
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)[reply]

I did a thing, hopefully it's smoother now. --JBL (talk) 17:44, 3 July 2023 (UTC)[reply]
Thanks XOR'easter. 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)[reply]

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)[reply]

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)[reply]

Hi Jo-Jo Eumerus, 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)[reply]

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'easter (talk) 21:13, 24 July 2023 (UTC)[reply]

Thanks, I will try to run it down. --JBL (talk) 00:19, 26 July 2023 (UTC)[reply]

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)[reply]

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'easter (talk) 18:27, 25 July 2023 (UTC)[reply]

Last sentence before the paragraph break. Jo-Jo Eumerus (talk) 19:15, 25 July 2023 (UTC)[reply]

In which paragraph? XOR'easter (talk) 19:58, 25 July 2023 (UTC)[reply]

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)[reply]

Support by Iry-Hor

Jo-Jo Eumerus, Gog the Mild 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)[reply]

Thanks Iry-Hor. At the moment the article is in dire need of reviewers. Perhaps Daniel Case 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)[reply]

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)[reply]

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)[reply]

Hi Shapeyness, 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)[reply]

On first reading, I found Each one is an infinite extension of a finite symmetric group, the group of permutations (rearrangements) of a finite set. 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)[reply]
This is far better yes, and makes even some of the unchanged sentences a lot clearer! Shapeyness (talk) 13:22, 26 July 2023 (UTC)[reply]
In addition to their geometric description... what geometric description? Is this talking about the idea that they are mathematical structures that describe the symmetries of the number line and the regular triangular tiling of the plane?
Yes, that was the intention. --JBL (talk) 21:16, 25 July 2023 (UTC)[reply]
A small signpost-y sentence might also be useful here, e.g. Affine symmetric groups can be defined in a number of ways.
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)[reply]
The relations of the definition are a system of equations satisfied by those elements that imply all of the other equations they satisfy. 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)[reply]
Would something like The relations used in the definition are a system of equations satisfied by elements of the group from which all other relations between group elements can be inferred work? Just a suggestion and probably not the most elegant way of putting it. Shapeyness (talk) 13:29, 26 July 2023 (UTC)[reply]
(The second and third relation are sometimes called the braid relations.) 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)[reply]
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)[reply]
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'easter (talk) 18:21, 25 July 2023 (UTC)[reply]
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)[reply]
Ok, thanks, just wanted to make sure. Shapeyness (talk) 22:18, 25 July 2023 (UTC)[reply]
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)[reply]
Ok, that's what I suspected, thanks. Shapeyness (talk) 22:13, 25 July 2023 (UTC)[reply]

Also, yep, should have read through the nomination statement more thoroughly, sorry about that! Shapeyness (talk) 13:41, 26 July 2023 (UTC)[reply]
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)[reply]

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.
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.
Connection between the geometric and combinatorial definitions: Perhaps this can also be signposted at the top of the section following The affine symmetric group contains the finite symmetric group of permutations on n elements as both a subgroup and a quotient group., e.g. something like The finite symmetric group provides a method for directly translating between the combinatorial and geometric definitions of the affine symmetric group. This just helps assure the reader this subsection doesn't belong to the previous definitions section.
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

@@ Line 33: / Line 33: @@
 ::::Last sentence before the paragraph break. [[User:Jo-Jo Eumerus|Jo-Jo Eumerus]] ([[User talk:Jo-Jo Eumerus|talk]]) 19:15, 25 July 2023 (UTC)
 :::::In which paragraph? [[User:XOR&#39;easter|XOR&#39;easter]] ([[User talk:XOR&#39;easter|talk]]) 19:58, 25 July 2023 (UTC)
+::::::{{tq|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 {{tq|For example, a portion of the matrix for the affine permutation  is shown in the figure}} for example. [[User:Jo-Jo Eumerus|Jo-Jo Eumerus]] ([[User talk:Jo-Jo Eumerus|talk]]) 16:42, 26 July 2023 (UTC)
 ====Support by Iry-Hor====