97:
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
587:
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
Knowledge 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.
258:, 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.
2592:. 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.
1016:). 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.
502:): 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
2596:
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.
533:: 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 Knowledge audience the application of
988:
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
586:
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
268:
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
423:
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
2617:
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,
139:
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
2595:
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
2587:
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
1382:
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
350:
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"
2558:
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
1570:
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
1019:
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
1011:
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
2530:
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
802:
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
798:
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.
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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
1131:
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
238:
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? Knowledge 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
1020:
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). --
1383:
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.
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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
1136:
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
701:
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!).
96:
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
269:
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?)
779:(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. --
775:
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.
1674:. 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
2566:
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
125:, 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.
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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
438:
It is not however accepted
Knowledge style to make readers go backward in the paragraph to find the citations. Maybe that's accepted style in mathematics textbooks, but Knowledge is notionally supposed to work for a general audience as well.
2118:+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). --
2206:
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
697:
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.
1631:
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.
1203:
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.
1703:
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.
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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.
2031:
402:
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
614:
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.
261:
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
1502:
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.
524:
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
2648:, 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?
2554:
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
218:
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.
2140:
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.
767:
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,
2188:
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?
1104:
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.
1121:
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.
658:
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.
336:
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
Knowledge.
318:
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. --
901:
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). --
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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.
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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.
506:: 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
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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
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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. --
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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. --
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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.
144:, 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.
887:
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.
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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. --
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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
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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).
2477:
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. --
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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.
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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
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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". --
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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. --
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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.
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The finite symmetric group provides a method for directly translating between the combinatorial and geometric definitions of the affine symmetric group
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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. --
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Yes, "Construction of root systems". The sentence "For Δ take " gives (explicitly, but in a different notation) the fact that the generators are
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Section 4.3 of
Humphreys talks about alcoves that are not equilaterial triangles, but it doesn't say anything specific about higher dimensions.
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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).
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Thanks for your additional comments, and sorry for the delay in responding to them. With respect to the three sentences you quoted above (in
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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.
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The relations of the definition are a system of equations satisfied by those elements that imply all of the other equations they satisfy.
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Connection between the geometric and combinatorial definitions: Perhaps this can also be signposted at the top of the section following
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This has changed the reference numbers; consequently I have adjusted all the footnote numbers referred to in your comments below. --
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The affine symmetric group contains the finite symmetric group of permutations on n elements as both a subgroup and a quotient group.
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happy to report that Green offers a substantive assertion on this topic, and so I was able to expand the section (very slightly). --
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I think the template at the top is somewhat misleading - the linked article was derived from our article, not the other way around.
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Footnotes 13 and 60 are to whole books, Shi (1986) and Lehrer & Taylor (2009) respectively. Could these be made more specific?
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Coxeter groups have a number of special properties not shared by all groups. These include that they have decidable word problem
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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) ). --
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Each one is an infinite extension of a finite symmetric group, the group of permutations (rearrangements) of a finite set.
529:, expecting readers to be able to fill in the details themselves. (See, for example, the treatment of Figures 6 and 9 in
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I've given it another read. The only thing I've found that ought to be cited is in the "Geometric definition" subsection:
1842:
Oh dear I've now made the tracking of different versions hopeless :(. I agree that finer pagination would be better: in
457:. They're just not. Most of the are going to nope out exactly nine words into the article, when it says "mathematical".
1344:, still mulling one over. I will try to get to it in the near future, and will ping Shapeyness when I do -- thanks. --
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a math degree that they've taken a group theory course will be capable of looking a sentence backwards for a footnote.
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This is the same as the equation numbered (8.30), up to notational differences between our article and the source. --
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Yeah, sadly there seems to be little discussion of the history of this object particularly in reliable sources :(. --
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mathematical structures that describe the symmetries of the number line and the regular triangular tiling of the plane
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The referenced section (on journal page 67 / PDF page 7) is titled "Tabloids and AMBC" and begins "Given a partition
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I reviewed this article some time ago and had not noticed it being here owing to being busy outside of Knowledge. I
974:. Perhaps a line could be added about what exactly Shi contributed beyond what was already known at that point.
194:. If you have done the donkey work, do feel free to finish it off here. And/or do a standard source review. :-)
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2211:). I've added a separate reference to an earlier spot in BB for the definition of "parabolic subgroup". --
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21: What chapter does this refer to? "Construction of root systems"? That does not seem to source anything.
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disappointing: after considerable effort, I was able to find sources that explicitly detail the claim that
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the various copies of this reference have been replaced with the more precise references 7, 8, 18, 26. --
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is used in the singular but is meant in the plural (the statements apply to hypercubes of any dimension).
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1326:. Have you addressed all of Shapeyness's comments? If so, could you ping them to let them know? Thanks.
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1234:. This just helps assure the reader this subsection doesn't belong to the previous definitions section.
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846:, pushing the technicality into a footnote (where it can be expanded on without burdening the text). --
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here as this clearly wouldn't be unverifiable, but taking a cautious approach for FA also makes sense.
677:, thanks very much for your comments! I will respond inline below (first batch now, more to come). --
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20: What chapter does this refer to? "Affine Weyl group"? A mathematician probably needs to check it.
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OK, I see I've already commented but there was a request for a source review, so I'll put one here on
1453:... I don't know if that's standard phrasing for the area, but it looks a bit like a word is missing.
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Yeah you're absolutely right, the flow is funny there. That is a good suggestion, I will adjust. --
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in this case is dubious. Between the two sources, however, I think it is now adequately sourced. --
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Spot-check but really needs replication from someone who actually knows this kind of mathematics:
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I can confirm that there is no plagiarism nor close paraphrasing from the sources used, though.
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Also, yep, should have read through the nomination statement more thoroughly, sorry about that!
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London Mathematical Society 2.1 (1987): 42-55.) page 55 specifically, which is available here:
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The history section is quite short, does it cover all the major points from reliable sources?
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has been promoted, but there may be a delay in bot processing of the close. Please see
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The affine symmetric group is closely related to a variety of other mathematical objects
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Just a thought: the "History" section currently says that the combinatorial description
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847:
812:
780:
769:
745:
703:
678:
611:
538:
534:
526:
511:
503:
407:
For example, a portion of the matrix for the affine permutation is shown in the figure
319:
305:
273:
239:
paragraphs and unsourced sentences, of which there are some. As well as "we" language.
177:
159:
122:
102:
81:
716:
This is far better yes, and makes even some of the unchanged sentences a lot clearer!
632:
Between administrative work and the content I'm working on, I have a lot to do, but I
1179:
2026:{\displaystyle \{(1,-1,0,\ldots ,0),(0,1,-1,\ldots ,0),\ldots ,(0,\ldots ,0,1,-1)\}}
2663:
2645:
2612:
2597:
2574:
1277:
1258:
1143:
1082:
I will attempt another read through soon and give any more comments if I have them.
603:
588:
98:
2495:, have you addressed all of Jo-Jo's comments? If so, could you ping them? Thanks.
510:). The other one is a bit more subtle and I will get back to you on it later. --
351:
language is generally more textbook-like than Knowledge house style prefers, but
2700:
57:
1868:
Specifically the first sub-section "The affine Weyl group" (pages 118–119). --
803:
expressions for the same quantity). The confusing part is the assertion that
829:
work? Just a suggestion and probably not the most elegant way of putting it.
453:
A "general audience" is not going to be reading multiple sections down into
2258:
36: The equation for exp ˜S (x; q) is somewhat different from the source.
2573:
Footnote 64. Can't access right now but I can try these two again later.
867:(The second and third relation are sometimes called the braid relations.)
737:
what geometric description? Is this talking about the idea that they are
508:
Mathematical literacy may be necessary to follow a "routine" calculation
355:
seems to have gotten the last of it. I did a find-in-page just now for
1678:
sometimes we are citing page numbers and other times entire chapters.
530:
1810:" but it seems clear I have the wrong page-number. I will fix it. --
1479:
to be articles; the treatment here just seems a little asymmetrical.
1265:
should probably also be linked at first occurrence in the body, also
924:
As it happens, I mentioned the brevity of the history section during
288:
However for higher dimensions, the alcoves are not regular simplices.
1302:
Thanks both. I've linked rep thry in the first use in the body. --
1237:
This is a nice suggestion, thanks -- I've added a sentence there. --
140:
errors can catch the best of us. One quick note about the prose: in
39:
Subsequent comments should be made on the article's talk page or in
2208:
1273:
1254:
606:. At the moment the article is in dire need of reviewers. Perhaps
1701:
I've added a separate reference to support the parenthetical.
463:
The citation should be added close to the material it supports
1505:
Then the affine symmetric group can be realized geometrically
1734:
Originally that footnote was attached to the statement "For
760:
A small signpost-y sentence might also be useful here, e.g.
1431:
Thanks for your comments; I've responded to them below. --
928:, and it seems there's just not the material to expand it.
762:
Affine symmetric groups can be defined in a number of ways
2230:
33: Maybe not ("left"?), needs a mathematician to check.
2547:
2440:
2414:
2354:
55: Is the equation supposed to be B2 from the source?
1843:
1671:
1365:
I think I've responded to all of your comments now. --
843:
499:
352:
173:
65:
2385:
Really? The cited section is only 1.5 pages long. --
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48: I am not seeing the definition of "tabloid" here.
2315:
2070:
2039:
1896:
1796:
1770:
1740:
1730:
4: What piece of information comes from this source?
1577:
1513:
610:
might be enticed into writing one? Or perhaps you or
469:
do not go into detail at all; the recently beefed-up
2590:
https://math.ecnu.edu.cn/~jyshi/myart/1987JLMalc.pdf
475:
must be cited no later than the end of the paragraph
1790:it consists of two nodes joined by an edge labeled
2327:
2102:
2052:
2025:
1802:
1782:
1752:
1590:
1535:
2720:The above discussion is preserved as an archive.
2060:means the vector that has a single 1 in position
43:. No further edits should be made to this page.
2726:No further edits should be made to this page.
2699:template in place on the talk page until the
2473:Will look at this list soon, but just noting
735:In addition to their geometric description...
29:The following is an archived discussion of a
8:
2020:
1897:
265:What sentences do you believe are unsourced?
2436:77: This probably need a finer pagination.
2410:69: This probably need a finer pagination.
2382:62: This probably need a finer pagination.
1545:Other aspects of the affine symmetric group
2404:64: Maybe, needs a mathematician to check.
2376:61: This probably need a finer pagination.
2252:34: Maybe, needs a mathematician to check.
2160:25: Maybe, needs a mathematician to check.
1839:7: This probably need a finer pagination.
1651:Concerns addressed; not much else to say.
1549:Other aspects of an affine symmetric group
1482:Yeah I dunno -- maybe some day I'll write
373:Last sentence before the paragraph break.
41:Knowledge talk:Featured article candidates
2413:Done (as footnotes 69, 70, 73, 75, 76 in
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2075:
2069:
2044:
2038:
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1516:
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2233:Specifically this is Definition 3.3. --
2192:I have removed the repeated footnote. --
2546:Brirush's comments below correspond to
2474:
1684:3: I am not seeing "braid relations"?
1556:
1552:
1548:
1544:
1507:, for example, might be better with a
1504:
1450:
1062:Oh no need to apologize, I'm sure! --
971:
956:Ok, thanks, just wanted to make sure.
507:
474:
462:
406:
401:
304:Thanks, I will try to run it down. --
287:
2618:so I will replace it in a moment. --
2110:is a vector that has a 1 in position
2033:. (The specific translation is that
1034:Ok, that's what I suspected, thanks.
18:Knowledge:Featured article candidates
7:
2166:29: I don't see the equation there?
1760:, its Coxeter–Dynkin diagram is the
1174:I do think there is an argument for
359:as a whole word and got no results.
1160:Yes, it's a frustrating situation.
2439:Done (as references 82, 83, 85 in
2335:, a tabloid of shape λ is ..." --
1797:
1557:The affine symmetric groups are...
176:, hopefully it's smoother now. --
24:
2328:{\displaystyle \lambda \vdash n}
1536:{\displaystyle {\tilde {S}}_{n}}
744:Yes, that was the intention. --
2570:Footnote 34. Can't access this.
1486:, for now I've unlinked it. --
972:was expanded upon by Shi (1986)
2280:37: Can't access this source.
2137:22: Can't access this source.
2017:
1984:
1972:
1939:
1933:
1900:
1521:
1407:I'm close to supporting this.
1:
2103:{\displaystyle e_{i}-e_{i+1}}
1712:) 17:37, 14 August 2023 (UTC)
1086:Ok, few more comments below.
2064:and all other entries 0, so
1818:) 21:23, 9 August 2023 (UTC)
1563:As with the previous point,
2713:21:43, 22 August 2023 (UTC)
2694:featured article candidates
2672:19:58, 22 August 2023 (UTC)
2658:12:45, 22 August 2023 (UTC)
2628:17:13, 22 August 2023 (UTC)
2606:20:18, 21 August 2023 (UTC)
2583:16:59, 21 August 2023 (UTC)
2541:20:09, 16 August 2023 (UTC)
2519:19:16, 16 August 2023 (UTC)
2505:17:13, 16 August 2023 (UTC)
2453:17:59, 22 August 2023 (UTC)
2427:20:06, 16 August 2023 (UTC)
2395:20:06, 16 August 2023 (UTC)
2367:19:50, 13 August 2023 (UTC)
2243:19:26, 16 August 2023 (UTC)
2221:19:26, 16 August 2023 (UTC)
2128:17:55, 14 August 2023 (UTC)
1878:17:55, 14 August 2023 (UTC)
1856:17:24, 22 August 2023 (UTC)
1830:19:26, 16 August 2023 (UTC)
1724:19:26, 16 August 2023 (UTC)
1272:Good comments. I've linked
31:featured article nomination
2743:
2487:17:44, 8 August 2023 (UTC)
2469:15:47, 8 August 2023 (UTC)
2345:21:23, 9 August 2023 (UTC)
2297:21:23, 9 August 2023 (UTC)
2287:22, see response above. --
2271:21:23, 9 August 2023 (UTC)
2202:21:27, 9 August 2023 (UTC)
2179:21:23, 9 August 2023 (UTC)
2151:21:23, 9 August 2023 (UTC)
1697:21:23, 9 August 2023 (UTC)
1661:23:15, 5 August 2023 (UTC)
1642:23:10, 5 August 2023 (UTC)
1622:17:53, 2 August 2023 (UTC)
1608:23:18, 1 August 2023 (UTC)
1496:23:18, 1 August 2023 (UTC)
1466:23:18, 1 August 2023 (UTC)
1441:23:18, 1 August 2023 (UTC)
1417:18:03, 1 August 2023 (UTC)
1393:20:01, 5 August 2023 (UTC)
1375:18:36, 4 August 2023 (UTC)
1354:22:58, 1 August 2023 (UTC)
1336:11:22, 1 August 2023 (UTC)
1214:21:17, 5 August 2023 (UTC)
1192:20:01, 5 August 2023 (UTC)
1170:21:42, 4 August 2023 (UTC)
1156:18:36, 4 August 2023 (UTC)
999:21:15, 5 August 2023 (UTC)
984:21:46, 4 August 2023 (UTC)
693:On first reading, I found
547:18:36, 4 August 2023 (UTC)
520:00:59, 2 August 2023 (UTC)
491:17:30, 1 August 2023 (UTC)
449:09:04, 1 August 2023 (UTC)
2283:Same article as footnote
1543:inserted to be explicit.
1312:17:20, 30 July 2023 (UTC)
1298:17:02, 28 July 2023 (UTC)
1247:17:56, 30 July 2023 (UTC)
1115:17:56, 30 July 2023 (UTC)
1096:15:32, 26 July 2023 (UTC)
1072:17:56, 30 July 2023 (UTC)
1058:13:41, 26 July 2023 (UTC)
1044:22:13, 25 July 2023 (UTC)
1030:21:16, 25 July 2023 (UTC)
966:22:18, 25 July 2023 (UTC)
952:21:16, 25 July 2023 (UTC)
938:18:21, 25 July 2023 (UTC)
911:17:56, 30 July 2023 (UTC)
897:13:38, 26 July 2023 (UTC)
883:21:16, 25 July 2023 (UTC)
869:Why is this in brackets?
856:17:56, 30 July 2023 (UTC)
839:13:29, 26 July 2023 (UTC)
821:00:32, 26 July 2023 (UTC)
789:21:16, 25 July 2023 (UTC)
754:21:16, 25 July 2023 (UTC)
726:13:22, 26 July 2023 (UTC)
712:00:24, 26 July 2023 (UTC)
687:21:16, 25 July 2023 (UTC)
668:16:04, 25 July 2023 (UTC)
646:05:05, 26 July 2023 (UTC)
625:20:56, 24 July 2023 (UTC)
597:16:19, 24 July 2023 (UTC)
467:Featured article criteria
434:21:51, 31 July 2023 (UTC)
419:16:42, 26 July 2023 (UTC)
397:19:58, 25 July 2023 (UTC)
383:19:15, 25 July 2023 (UTC)
369:18:27, 25 July 2023 (UTC)
346:07:17, 25 July 2023 (UTC)
328:17:20, 30 July 2023 (UTC)
314:00:19, 26 July 2023 (UTC)
300:21:13, 24 July 2023 (UTC)
282:17:47, 24 July 2023 (UTC)
249:09:02, 24 July 2023 (UTC)
229:12:16, 21 July 2023 (UTC)
135:18:00, 30 June 2023 (UTC)
111:17:20, 30 June 2023 (UTC)
90:17:20, 30 June 2023 (UTC)
2723:Please do not modify it.
1449:A minor phrasing issue:
204:19:01, 4 July 2023 (UTC)
186:17:44, 3 July 2023 (UTC)
168:23:13, 2 July 2023 (UTC)
154:21:39, 2 July 2023 (UTC)
36:Please do not modify it.
1803:{\displaystyle \infty }
1753:{\displaystyle n\geq 3}
1269:from slightly later on
2329:
2104:
2054:
2027:
1804:
1784:
1754:
1592:
1571:standard the same way
1537:
1403:Comments by XOR'easter
1230:, e.g. something like
654:Comments by Shapeyness
629:Since I was pinged ...
455:affine symmetric group
142:§ Algebraic definition
73:Affine symmetric group
2330:
2114:and a −1 in position
2105:
2055:
2053:{\displaystyle e_{i}}
2028:
1805:
1785:
1755:
1593:
1591:{\displaystyle S_{n}}
1565:Hyperoctahedral group
1538:
1286:Representation theory
1263:representation theory
825:Would something like
471:Good article criteria
116:First-time nomination
2313:
2068:
2037:
1894:
1794:
1768:
1738:
1612:Good edits. Thanks.
1575:
1511:
1402:
477:but do not rule out
387:In which paragraph?
234:Some random comments
101:before submission.
1783:{\displaystyle n=2}
1547:might be better as
2325:
2100:
2050:
2023:
1800:
1780:
1764:-cycle, while for
1750:
1588:
1533:
1261:should be linked,
571:Support by Iry-Hor
2531:comments now.) --
1524:
1282:Cartan subalgebra
1267:Cartan subalgebra
495:Hi Jo-Jo Eumerus,
473:say that content
92:
64:) 22 August 2023
2734:
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2689:, and leave the
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214:Coordinator note
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48:The article was
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80:Nominator(s):
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34:
22:
21:
20:
12:
11:
5:
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2703:goes through.
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1484:Core partition
1473:Core partition
1470:
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1400:
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773:
770:MOS:LEADLENGTH
765:
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732:
731:
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690:
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655:
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651:
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272:All the best,
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2526:Jo-Jo Eumerus
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2511:Jo-Jo Eumerus
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2461:Jo-Jo Eumerus
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1667:Source review
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1134:Jo-Jo Eumerus
1128:
1123:
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844:I did a thing
842:
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576:Jo-Jo Eumerus
570:
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441:Jo-Jo Eumerus
437:
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411:Jo-Jo Eumerus
409:for example.
408:
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375:Jo-Jo Eumerus
372:
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335:
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256:Jo-Jo Eumerus
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233:
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19:
2722:
2719:
2705:Gog the Mild
2679:Closing note
2678:
2650:Gog the Mild
2563:that's fine.
2555:projection).
2548:this version
2545:
2497:Gog the Mild
2441:this version
2433:
2415:this version
2407:
2401:
2379:
2373:
2351:
2303:
2284:
2277:
2255:
2249:
2227:
2185:
2163:
2157:
2134:
2115:
2111:
2061:
1884:
1862:
1844:this version
1836:
1733:
1702:
1679:
1675:
1672:this version
1648:
1555:ought to be
1476:
1406:
1379:
1342:Gog the Mild
1328:Gog the Mild
1278:Bruhat order
1259:Bruhat order
1085:
808:
804:
657:
633:
617:Gog the Mild
583:
580:Gog the Mild
574:
500:this comment
478:
356:
237:
221:Gog the Mild
217:
196:Gog the Mild
127:Gog the Mild
95:
54:Gog the Mild
49:
47:
35:
28:
638:Daniel Case
608:Daniel Case
459:WP:CITEFOOT
174:did a thing
1653:XOR'easter
1614:XOR'easter
1551:. Perhaps
1427:XOR'easter
1409:XOR'easter
1385:Shapeyness
1361:Shapeyness
1290:XOR'easter
1199:Shapeyness
1184:Shapeyness
1162:XOR'easter
1127:Shapeyness
1088:Shapeyness
1050:Shapeyness
1036:Shapeyness
1014:WP:ONEDOWN
976:XOR'easter
958:Shapeyness
930:XOR'easter
889:Shapeyness
831:Shapeyness
718:Shapeyness
675:Shapeyness
660:Shapeyness
531:this paper
483:XOR'easter
426:XOR'easter
403:simplices.
389:XOR'easter
361:XOR'easter
292:XOR'easter
192:XOR'easter
146:XOR'easter
2687:WP:FAC/ar
2683:candidate
2493:JayBeeEll
1178:and even
353:this edit
262:through.)
123:JayBeeEll
2681:: This
1253:I think
584:strongly
479:earlier.
50:promoted
2664:Brirush
2646:Brirush
2613:Brirush
2598:Brirush
2575:Brirush
1649:Support
1560:tweaks.
1380:Support
1176:WP:CALC
1146::(. --
1139:WP:CALC
604:Iry-Hor
602:Thanks
589:Iry-Hor
535:WP:CALC
527:WP:CALC
504:WP:CALC
190:Thanks
99:Iry-Hor
2551:--JBL
2417:). --
1280:, and
1180:WP:IAR
465:; the
58:FACBot
2567:good.
2443:). --
2209:coset
1477:ought
1274:coset
1255:coset
1144:WP:OR
809:every
805:every
461:says
16:<
2709:talk
2668:talk
2654:talk
2624:talk
2602:talk
2579:talk
2537:talk
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2198:talk
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1826:talk
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1657:talk
1638:talk
1618:talk
1604:talk
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1462:talk
1437:talk
1413:talk
1389:talk
1371:talk
1350:talk
1332:talk
1308:talk
1294:talk
1257:and
1243:talk
1210:talk
1188:talk
1166:talk
1152:talk
1111:talk
1092:talk
1068:talk
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1040:talk
1026:talk
995:talk
980:talk
962:talk
948:talk
934:talk
907:talk
893:talk
879:talk
852:talk
835:talk
817:talk
785:talk
750:talk
722:talk
708:talk
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664:talk
642:talk
634:will
621:talk
593:talk
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405:and
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62:talk
56:via
2701:bot
2644:Hi
2620:JBL
2533:JBL
2491:Hi
2479:JBL
2445:JBL
2419:JBL
2387:JBL
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2194:JBL
2171:JBL
2143:JBL
2120:JBL
1870:JBL
1848:JBL
1822:JBL
1812:JBL
1716:JBL
1706:JBL
1689:JBL
1676:but
1634:JBL
1600:JBL
1488:JBL
1458:JBL
1433:JBL
1367:JBL
1346:JBL
1340:Hi
1324:JBL
1322:Hi
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1206:JBL
1148:JBL
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991:JBL
944:JBL
903:JBL
875:JBL
848:JBL
813:JBL
781:JBL
746:JBL
704:JBL
679:JBL
673:Hi
612:JBL
539:JBL
512:JBL
320:JBL
306:JBL
274:JBL
254:Hi
178:JBL
160:JBL
121:Hi
103:JBL
82:JBL
52:by
2711:)
2697:}}
2691:{{
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