Talk:WikiJournal of Science/Affine symmetric group

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WikiJournal of Science
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WikiJournal of Science is an open-access, free-to-publish, Wikipedia-integrated academic journal for science, mathematics, engineering and technology topics. WJS WikiJSci Wiki.J.Sci. WikiJSci WikiSci WikiScience Wikiscience Wikijournal of Science Wikiversity Journal of Science WikiJournal Science Wikipedia Science Wikipedia science journal STEM Science Mathematics Engineering Technology Free to publish Open access Open-access Non-profit online journal Public peer review

<meta name='citation_doi' value='10.15347/WJS/2021.003'>

Article information

Author: Joel Brewster Lewis[a][i] 

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  1. George Washington University


Plagiarism check

Pass. Report from WMF copyvios tool: only similarities are common phrases or reference names (e.g. "The generalized Robinson–Schensted algorithm on the affine Weyl group of type"). T.Shafee(Evo﹠Evo)talk 04:53, 12 June 2020 (UTC)Reply

Peer review 1

Review by Martina Balagovic , Newcastle University
These assessment comments were submitted on , and refer to this previous version of the article

The article is correct, clearly written, interesting, and covers a fundamental topic in pure mathematics in a clear and accessible way. It should be included in the journal and in Wikipedia.

A few minor suggestions are listed below.

1. Algebraic definition - I personally prefer the notation s1, . . . , sn−1, s0 for the generators (so, swapping sn for s0). This doesn’t really matter - the author clearly states the indices are modulo n - but it avoids the possibility of silly errors (as s1, . . . , sn−1, sn are commonly generators of Sn+1), generalises to other types, and is more common in representation theory, where s0 matches the notation for the “affine” generators e0, f0, h0. Indeed, later in the article the notation switches to s0 so it would be good to be consistent.

2. Combinatorial definition [0, 2, 3, . . . , n − 2, n − 1, . . . , n + 1] - should that be [0, 2, 3, . . . , n − 2, n − 1, n + 1]?

3. Connection between the geometric and combinatorial definitions - more can be said - Λ is not just an Abelian group, it is a free abelian group with n − 1 nice explicit generators. These generators have interpretations in representation theory, so even if this is not explored here (as it might not be the best place), it might make sense to say that Λ is a free abelian group and to list the generators.

4. Extended affine symmetric group - it might be good to mention the in-clusion of the affine symmetric group into the extended affine symmetric group.

5. It would be good to add a paragraph about the connection with the affine Lie algebras of type A.


Many thanks to the referee for this thoughtful and constructive report. With respect to the individual comments:

  1. Yes, absolutely. I have changed s_n to s_0 in the section "Algebraic definition" and one or two other places.
  2. Indeed! I have corrected the error.
  3. Yes, absolutely. I have added this description at the end of the section "As a quotient".
  4. I am not sure that I understand this comment correctly, and would welcome elucidation. (Currently, the section begins with the statement that the affine symmetric group is a subgroup of the extended affine symmetric group.)
  5. I have attempted to add a short section about this (here).

Thanks again to the referee for this report. Joel Brewster Lewis (discusscontribs) 02:53, 6 February 2021 (UTC)Reply

Peer review 2

Review by Kyle Petersen , DePaul University
These assessment comments were submitted on , and refer to this previous version of the article

This article is a fantastic summary of many key features of the affine symmetric group! While I suppose there are other topics that might have been included (e.g., Reiner's generating function for length and descents, or more about geometry of the Tits cone and its quotient known as the Steinberg torus) the choice of further topics is a matter of taste, and I find no fault with the topics chosen. The connection with juggling patterns is really fun, and perhaps a bit more on that model would be nice (perhaps with an illustration from Ehrenborg and Readdy). I think this article will be a great resource for students and others learning about the affine symmetric group for the first time.


Thanks very much to the referee for these suggestions. I have added a discussion of Reiner's generating function to the section on inversions and descents, where it fits nicely. I agree that the discussion of the connection with juggling is wonderful, and that it was not given its due in the first version. I have expanded the section significantly, including some illustrations. --Joel Brewster Lewis (discusscontribs) 18:54, 24 February 2021 (UTC)Reply

Peer review 3

Review by Brant Jones , James Madison University
These assessment comments were submitted on , and refer to this previous version of the article

This is an accessible article with a very nice balance of breadth and depth on a subject that can be difficult to contextualize with its varied connections to topics in algebra, combinatorics, and geometry.

While optional, a couple of fundamental nuts-and-bolts issues that might be helpful to mention or introduce for researchers wanting to quickly get up to speed are:

1. How inversion / left-vs-right actions work in the various setups. Algebraically, one can multiply by Coxeter generators on the left or right and inversion switches these by reversing the factors in a product of generators, since the generators are all involutions. Combinatorially, one can act with a generator by interchanging the values of two entries or, dually, by interchanging their positions. Similarly, one can act in the geometric setting by reflecting an alcove through the fixed hyperplanes or, instead, across one of its local bounding walls to a geometrically adjacent alcove. Even if you don't want to get into all this in the text, it might be useful to label the geometric picture for n = 3 with both actions.

2. The parabolic decomposition as a way to simplify the "essential data" required by the various setups. Each affine permutation can be viewed as a finite permutation together with a minimal length coset representative. For example, in the combinatorial setup the minimal length coset representative corresponds to the choice of entries in the base window, while the finite permutation determines their relative order.

Combinatorial interpretations for the minimal length coset representatives include abacus diagrams (as you describe), but also core partitions and bounded partitions, which allow different statistics and properties to be read from these objects. See e.g. "A bijection on core partitions and a parabolic quotient of the affine symmetric group" by Berg, Jones, and Vazirani in Journal of Combinatorial Theory, Series A 116 (2009) 1344–1360.

Also, there is another Hanusa & Jones paper "Abacus models for parabolic quotients of affine Weyl groups" in Journal of Algebra Volume 361, 1 July 2012, Pages 134-162 which generalizes the affine symmetric group combinatorics of these minimal length coset representatives to the other affine Weyl groups.

Citation note:

In the "Fully commutative elements and pattern avoidance" section, you should cite

"On 321-avoiding permutations in affine Weyl groups" by R.M. Green in J. Algebraic Combin., 15 (3) (2002), pp. 241-252

where you currently say "In (Hanusa & Jones 2010), this result was extended to affine permutations..." (The Hanusa & Jones paper you cite is primarily concerned with enumerating the fully commutative elements by length.)

Very minor notes:

1. It seems you should add a vertical window boundary between the top two elements (separating 2 from 0) in the figure of the "Representation as matrices" section.

2. I don't think you've mentioned s_n when you say "excluding the simple reflection " in the "Relationship to the finite symmetric group/As a subgroup" section.


Thanks to the referee for this thorough report! Responding to the various suggestions point-by-point:

  • Yes, I agree, this is a very good point. (Indeed, I spent a lot of time in the early stages of writing making sure all the conventions were internally consistent -- so it is odd not to mention the alternatives at all!) I have added a short section on this, including the dual labeling of the alcoves.
  • I have added a mention of alternative combinatorial models (core partitions and bounded partitions) in the section on parabolic subgroups and representatives, referencing Lapointe-Morse and Berg-Jones-Vazirani.
  • I have added a brief section on the combinatorial models of other affine Coxeter groups (the "George groups") and included a mention of parabolic quotients there.
  • Thank you for the pointer to Green's paper -- I have corrected the attribution, and clarified the contribution of Hanusa and Jones 2010.
  • In order to maintain agreement with the surrounding text, I have cropped one column of the matrix, so that no additional vertical rulings are needed.
  • In the section on the algebraic definition, I assert that indices are taken modulo n. Then in several places (e.g., at the end of the section on the geometric definition) I make a similar reference to "the reflection ". I have significantly reduced the appearance of in response to a comment from the first referee; I am inclined not to go further in eliminating , but I could if it is still confusing.

Thanks again for these constructive comments! --Joel Brewster Lewis (discusscontribs) 20:09, 18 March 2021 (UTC)Reply

Editorial notes

Comments by Thomas Shafee ,
These editorial comments were submitted on , and refer to this previous version of the article

Below are some final notes from discussions with the other editors on structure and clarity:

Current sections 3 and 4 are well structured ("Definitions" and "Relationship to the finite symmetric group"), whereas the subsequent section have a very flat organisation. It might aid readability to also structure later sections a little more. Some tentative suggestions:

  • sections 5 to 8 -> "subgroups" / "substructures" ?
  • sections 9 to 12 -> "group actions" ?
  • sections 13 to 16 -> "Relationships" ?

Given the highly technical nature of the topic, a additional very short plain language / nontechnical summary would be very useful for a lay or non-mathematician reader to get a quick take-home overview. A couple of resources:

  • Gudi, Sai Krishna (2021-03-18). "Plain-Language Summaries: An Essential Component to Promote Knowledge Translation". Your Say. Retrieved 2021-03-21.
  • Salita, Joselita T. (2015-12-01). "Writing for lay audiences: A challenge for scientists". Medical Writing 24 (4): 183–189. doi:10.1179/2047480615Z.000000000320. ISSN 2047-4806. 

Thanks @Thomas Shafee: and the editorial board for your comments. I have introduced some additional section headings to the article (similar but not identical to your suggestions). Those additions were accompanied by some rearrangement of sections and the addition of guiding text. I also added a concrete example in the section on juggling patterns, to better connect the text with the figures.

I have rewritten the abstract to be less technical, but on rereading this editorial comment, it's not clear to me that's what was requested. Does the new version of the abstract satisfy the request, or would you like a lay summary separate from the article abstract?

Separately, I wonder if there is a house style for adding acknowledgements? --Joel Brewster Lewis (discusscontribs) 20:17, 27 March 2021 (UTC)Reply

@JayBeeEll: Thank you - the changes are great. It would be ideal, if possible, to have a short non-technical summary in addition to the abstract aimed at someone even less expert in the area. The aim is to have someone with as little expertise as possible, be able to understand some core feature of the topic (sometimes requiring synonyms for technical terms, analogies, and cautious simplification). There's no strict style for acknowledgements currently (short guidance here), though we may eventually more to include more structured data in future (STARDIT). T.Shafee(Evo﹠Evo)talk 03:27, 30 March 2021 (UTC)Reply
@Thomas Shafee: Thank you for your responses. I have added an acknowledgements section to the paper. Here is an attempt at a lay summary.

The affine symmetric group is a mathematical structure that describes the symmetries of the number line and the regular triangular tiling of the plane, as well as related higher dimensional objects. The individual elements of the affine symmetric group, which are called affine permutations, may also be interpreted as certain periodic rearrangements of the set of integers (..., −2, −1, 0, 1, 2, ...), as well as in purely algebraic terms. Unlike the symmetries of a single polygon or polyhedron and the collection of rearrangements of a finite set, there are infinitely many affine permutations. For this reason, the affine symmetric group provides an avenue to extend the study of symmetries of polyhedra or of groups of permutations to the infinite case. As a result, the affine symmetric group is of interest in several areas of mathematics, including combinatorics and representation theory. It also has connections with mathematical objects that were originally studied for independent reasons, such as complex reflection groups and juggling sequences.

Joel Brewster Lewis (discusscontribs) 21:18, 19 April 2021 (UTC)Reply
Excellent, thank you. These summaries are notoriously hard to write! I've made a suggested version below, but please check if any is over-simplified to the point of incorrectness:

Flat, straight-edged shapes (like trianges) or 3D ones (like pyramids) have only a finite number of symmetries. In contrast, the affine symmetric group is a way to mathematically describe all the symmetries possible of triangular tiles arranged on a infinitely large flat surface. As with many subjects in mathematics, it can also be thought of in a number ways: for example it also describes the symetries of the infinitely long number line, or the posible repeating arrangements of all integers (..., −2, −1, 0, 1, 2, ...). As a result, studying the affine symmetric group extends the study of symmetries of polyhedra or of groups of permutations to the infinite case. It also connects several topics in mathematics that were originally studied for independent reasons (such as combinatorics and representation theory) ranging from complex reflection groups to juggling sequences.

T.Shafee(Evo﹠Evo)talk 11:48, 20 April 2021 (UTC)Reply
@Thomas Shafee: Thanks for your edits! It is very hard to remember how much of technical language is jargon. I have tweaked your latest version slightly, and I think we are converging on something quite reasonable:

Flat, straight-edged shapes (like triangles) or 3D ones (like pyramids) have only a finite number of symmetries. In contrast, the affine symmetric group is a way to mathematically describe all the symmetries possible when an infinitely large flat surface is covered by triangular tiles. As with many subjects in mathematics, it can also be thought of in a number ways: for example, it also describes the symmetries of the infinitely long number line, or the possible arrangements of all integers (..., −2, −1, 0, 1, 2, ...) with certain repetitive patterns. As a result, studying the affine symmetric group extends the study of symmetries of polyhedra or of groups of permutations to the infinite case. It also connects several topics in mathematics that were originally studied for independent reasons, ranging from complex reflection groups to juggling sequences.

Joel Brewster Lewis (discusscontribs) 15:02, 20 April 2021 (UTC)Reply