Talk:Affine symmetric group

Beautiful article
Beautiful prose quality and attention to detail, here, with great illustration. I came across this in the GAN queue; even though I studied quite a bit of algebra, I am not in a position to determine whether the article meets the criteria. (I have no experience in combinatorics.) Good work!

The only thing I can offer is to suggest that there be some motivation for calling it $$\widetilde{S}_n$$ rather than another name. ($$\widetilde{S}_n$$ contains $${S}_n$$ as a subgroup, as you explain later, but not when it is first introduced.) But that may be hard to do in a way that preserves prose quality and flow, so feel free to ignore. Urve (talk) 23:25, 28 April 2021 (UTC)
 * Hi, thanks very much for the kind words! About the notation $$\widetilde{S}_n$$, first let me say why this is what I chose (which I think you understand already): by far the most common notation for the (usual, finite) symmetric group is $$S_n$$ (sometimes rendered with the S in other fonts, like $$\mathfrak{S}_n$$; other options are mentioned at Symmetric group, and I've even seen other random things like γ in some sources).  When there are "linear" (or finite) and "affine" versions of some object (like a Coxeter group), it seems very common to denote the affine one with a tilde on top (but again this is not universal; sometimes one sees a hat used, or for authors for whom the affine object is the "natural" one they might write W for the affine version and W0 for the linear version).  So the notation $$\widetilde{S}_n$$ is taking the intersection of these two common choices, and is widely in use in the literature.  I should be clear that these are my observations from reading sources -- I do not recall seeing anyone write down anything about the choice of notation.  I am reluctant to put something like this discussion (that I believe may not be stated in any reliable source) into the article.  However, perhaps your point is not that I should add text about why this notation is used, but instead that if I introduce the reader to the fact that the symmetric group is called S_n earlier, that by itself would be helpful?  Thanks again, JBL (talk) 16:36, 29 April 2021 (UTC)
 * That makes a lot of sense, . Since this is a/the traditional way of writing the affine symmetric group, and it's not explained in the literature explicitly, you are right to be uncomfortable to add it to the article -- it would of course be original. I don't think mentioning S_n being the symmetric group earlier would be too helpful. I suppose I was just confused, not having much experience in this particular area, as to why a tilde was used here - but that appears to be because of my lack of exposure. :) Best of luck, I imagine you'll pass GAN with flying colors and I look forward to seeing other content you contribute to the project. Urve  (talk) 02:05, 30 April 2021 (UTC)
 * That makes sense -- thanks again for your feedback! --JBL (talk) 15:53, 1 June 2021 (UTC)

Short exact sequence
In the "Relationship to the finite symmetric group" section, should there be some mention of the short exact sequence $$0\to \Z^{n-1}\to \widetilde{S}_n\to S_n\to 0$$ where the first group is the free abelian group of translations, affine permutations that don't permute the residues mod $$n$$ (with $$\Z^{n-1}$$ rather than $$\Z^n$$ because of the requirement that the shifts sum to zero) and the last one is the quotient from the "As a quotient" subsection? I mean, you sort of do already describe it, but without saying it's a short exact sequence and without stating that the kernel is actually a free abelian group. —David Eppstein (talk) 19:11, 31 May 2021 (UTC)
 * Freeness is mentioned, but it's snuck in the very last sentence of the section. Perhaps a short additional paragraph on that section, offering the short exact sequence as a summary.  Running off to lunch now, but I'll try to find a moment to think about it this afternoon. --JBL (talk) 15:53, 1 June 2021 (UTC)
 * What about this as a starting point? "The kernel $π$ is by definition the set of affine permutations whose underlying permutation is the identity. The window notations of such affine permutations are of the form $[1 - a_1 \cdot n, 2 - a_2 \cdot n, \ldots, n - a_n \cdot n]$, where $(a_1, a_2, \ldots, a_n)$ is an integer vector such that $a_1 + a_2 + \ldots + a_n = 0$, that is, where $(a_1, \ldots, a_n) \in \Lambda$. Geometrically, this kernel consists of the translations, that is, the isometries that shift the entire space $V$ without rotating or reflecting it. In an abuse of notation, the symbol $Λ$ is used in this article for all three of these sets (integer vectors in $V$, affine permutations with underlying permutation the identity, and translations); in all three settings, the natural group operation turns $Λ$ into an abelian group, generated freely by the $n − 1$ vectors $\{(1, -1, 0, \ldots, 0), (0, 1, -1, \ldots, 0), \ldots, (0, \ldots, 0, 1, -1)\}$. The relationship of the preceding paragraph may be expressed by the short exact sequence $0\to \Z^{n-1}\to \widetilde{S}_n\to S_n \mathop{\overset{\pi}{\to}} 0$. Here $\Z^{n - 1} \cong \Lambda$ is the free abelian group with $n − 1$ generators. |undefined" I am not at home so can't easily browse for citations, but I am sure they exist; I am a bit annoyed that neither paragraph has any at the moment.  (Also annoying: the Wikipedia LaTeX implementation seems to lose the correct spacing around \to if you use \overset ?  Bah.)--JBL (talk) 17:47, 1 June 2021 (UTC)
 * Looks ok to me. I added \mathop to the formula to space the \to better. —David Eppstein (talk) 18:52, 1 June 2021 (UTC)
 * Great, thanks! --JBL (talk) 20:33, 1 June 2021 (UTC)

True statements that I was not immediately able to source
Here I'm going to maintain a (hopefully short) list of things that are true but that I was not able to find sources for on first attempt: --JBL (talk) 11:30, 4 November 2021 (UTC)
 * "In terms of the geometric definition, this corresponds to the reflection across the plane $$x_i - x_{j} = k$$."
 * "The non-maximal parabolic subgroups of $$\widetilde{S}_n$$ are all isomorphic to parabolic subgroups of $$S_n$$, that is, to a Young subgroup $$S_{a_1} \times \cdots \times S_{a_k}$$ for some positive integers $$a_1, \ldots, a_k$$ with sum $n$."
 * One may define $$\widetilde{S}_n$$ to be the group of rigid transformations of $V$ that preserve the lattice $Λ$. (D'oh -- this is just false!)
 * In terms of the Coxeter generators of $$S_n$$, this can be written as $$ \pi(s_0) = s_1 s_2 \cdots s_{n - 2} s_{n - 1} s_{n - 2} \cdots s_2 s_1.$$
 * The relationship between the kernel, the affine symmetric group, and the image of $π$ may be expressed by the short exact sequence $$0\to \Z^{n-1}\to \widetilde{S}_n\mathop{\overset{\pi}{\to}} S_n\to 0$$. Here $$\Z^{n - 1} \cong \Lambda$$ is the kernel, a free abelian group with $n − 1$ generators.
 * The analogous combinatorial construction is to choose any subset $A$ of $$\mathbb{Z}$$ that contains one element from each conjugacy class modulo $n$ and whose elements sum to $$1 + 2 + \ldots + n$$; the subgroup $$(\widetilde{S}_n)_A$$ of $$\widetilde{S}_n$$ of affine permutations that stabilize $A$ is isomorphic to $$S_n$$.
 * Full details of the geometry for the infinite dihedral group: who are the roots, what is the alcove.
 * The sentence "However for higher dimensions, the alcoves are not regular simplices."

Affine symmetric group review
Dear I shall write here my comments rather than on your talk page as for some reason this caused your talk page to crash (it had to do with mathjax re-interpreting old stuff from your talk page for some reason). The article looks very good to me, informative and complete. I have a couple of questions/observations: More comments to come.Iry-Hor (talk) 09:44, 5 June 2023 (UTC)
 * There is a whole section on "Relationship to the finite symmetric group" and so I wondered why there was nothing on the relation to the group of braids on $$S^1$$ because as far as I can tell the affine symmetric group is a quotient of the braid group $$B_n(S^1)$$ of braids on $$S^1$$ under the relations $$s_i^2=Id>$$ for all i. Of course this is a simple observation, straightforward from the formal definition but I wonder why this isn't in the article. This quotient is compatible with the projection $$\pi$$ from $$\tilde{S}_n$$ to $$S_n$$, so is $$\pi$$ also the projection from $$B_n(S^1)$$ to $$S_n$$ (I think) where it is eminently simple to visualize. The relation to the braid group on the circle makes it easier (to me at least) to visualize the relations of the algebraic definition, in a manner similar to that given here.
 * In the section on the combinatorial definition, when you write $$\sum_{i=1}^n u(i) = 1+ 2+\cdots+n$$ I understand that this is easier for the general audience, but perhaps you could add $$ = \binom{n+1}{2}$$ or $$=\frac{n(n+1)}{2}$$ as these forms are commonly encountered in the literature when presenting this result.
 * As a Coxeter group (or as a quotient of the braid group), the word problem for the affine symmetric group is solvable. I think this should be stated somewhere.
 * In sentence "The generating function for these statistics over $$\widetilde{S}_{n}$$ simultaneously for all $$n$$ is" you could perhaps add the word "bivariate" in front of "generating" and you can wikilink to Generating_function
 * General question: let's say a reader wants to delve more into the subject, which book shall he/she go to ? Is there no general review book on the topic or are the historical item cited the best there is to be up to date on the notion ?Iry-Hor (talk) 09:44, 5 June 2023 (UTC)
 * Ref "Lusztig, George (1983), "Some examples of square integrable representations of semisimple p-adic groups", Trans. Amer. Math. Soc., 277: 623–653" is missing its DOI. Some other references are lacking DOI but have their MR numbers so it is fine by me. You might still get it pointed out at FAC though. Similarly none of the books have the publisher location listed but I think it is fine for FAC if nobody raises this in the sources review. It is actually better to have none than an inconsistent style with some that do and others that don't.
 * Two references are only arXiv preprint which might pose a problem at the FA source review. But these preprints have since been published. "Chmutov, Michael; Frieden, Gabriel; Kim, Dongkwan; Lewis, Joel Brewster; Yudovina, Elena (2018), Monodromy in Kazhdan-Lusztig cells in affine type A" is now published at Selecta Math. New Ser. 28, 67 (2022) and "Monodromy in Kazhdan-Lusztig cells in affine type A; Michael Chmutov, Joel Brewster Lewis, Pavlo Pylyavskyy" is in	Math. Annalen, 2022.

Technical comments for FAC: More comments to follow.Iry-Hor (talk) 11:46, 5 June 2023 (UTC)
 * I found a couple of duplinks : infinite dihedral group and identity element are wikilinked to several times in the main text.
 * All pictures must have alt text, which you can insert in the code with "| alt =" in the figure caption. This is mandatory for FA, see the MOS.
 * Note to anyone else interested in this discussion: it ended up taking place on my user talk-page. --JBL (talk) 17:09, 30 June 2023 (UTC)