User talk:Nbarth


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Dynkin diagram folding
Hi Nils von Barth,

I noticed you worked on the article/section Dynkin_diagram. I added some chart examples of these folding, although I can't say I did the "directed graph" aspect correctly. Also I used Coxeter notation, line for 3, labels above 3, rather than double/triple lines. I just added an arrow above the higher order lines. If you know better, please help. I'm surprised there's like a number of C~k versions by arrows >>, <<, and <>, based on foldings of different simply-laced higher graphs! Similarly F~4 can project from E~6 or E~7 with reversed arrows!? I have no confirmation what this means, but for the undirected Coxeter-Dynkin diagrams, it looks correct! Thanks for a look if you can help. Tom Ruen (talk) 05:48, 4 December 2010 (UTC)

p.s. On the directed graphs, Humphreys shows clearly directed arrows on a table on p96. So, either there's something that demands a direct direction, or when it was written (1990), folding wasn't firmly defined, and he didn't recognize alternate forms?! What do you think? Tom Ruen (talk) 06:40, 4 December 2010 (UTC)

p.p.s. I see the main limitation, is the Hn Coxeter group families are excluded from the Dynkin graphs, so if the charts are correct and can stay, they should be replaced in multiline format, and Hn cases removed. Tom Ruen (talk) 07:02, 4 December 2010 (UTC)


 * Hi Tom – thanks for your fabulous diagrams!
 * I’m not expert in this, so I did some reading – your diagrams look mostly correct, but you’re right, some changes would be in order, specifically:
 * Dynkin-only (eliminate Hn)
 * Affine: change the naming of the affine graphs – the different directions have different names
 * Add 1 more affine graph, and separate out the two ones mapping to “G~2” – I think they go to different graphs.
 * I elaborate below; the tables at Kac pp. 53–55 are our key reference!
 * To rephrase, the question you have is
 * “which way to direct the edges?”
 * The issue is that, for the affine diagrams, you get different directions (and your reasoning seems correct to me), not all from the first series (see below).
 * Thoughts follow (unindent for legibility):
 * Thoughts follow (unindent for legibility):


 * Firstly, there are several series of affine Dynkin diagrams; see affine Dynkin diagram, and tables on Kac pp. 53–55 (Infinite dimensional Lie algebras, Victor Kac) – I only just learned this.
 * These diagrams need to be labeled with superscripts (1, 2, or 3), as in $$A_{2l}^{(2)}$$ – the tildes correspond to the first series $$\tilde A_l = A_l^{(1)}$$ (the “extended” Dynkin diagrams), but there are some from the second series here in folding.
 * Direction isn’t much of a problem for the regular diagrams, but the Hn are not directed, so what exactly these maps mean is a bit unclear.
 * Further, there is a correct direction for the extended Dynkin diagrams; Humphreys is listing the first series:
 * (for reference, C~n is >-...-<, F~4 is -->-, and G~2 is <-).

The changes we get are:
 * By this reasoning, the 4 foldings A~2k–2 → C~k, D~k+2 → B~k+1, E~6 → F~4, and D~2k+1 → C~k correspond to maps in the second series – the arrows are wrong for the first series; you didn’t comment on the two directions for B…
 * It’s also not clear to me which way the arrow should go on “G~2” in E~7 → G~2, because the vertices on both sides of the multiple side are branch points, but I think the arrow should point to the center (the other way from E~6 → G~2); this maps to the third series.
 * The → H~k maps of (regular) diagrams don’t correspond to Lie algebras, which is a bit confusing or misleading in this context…
 * Also, the I2(8) diagram is an extended Dynkin diagram ($$A_2^{(2)}$$ – Kac page 55), so you may want to include it. (The other I2(p) are not affine Dynkin diagrams, though of course they are Coxeter ones.)

So, concretely, what I’d suggest is: I think this fixes it – all the diagrams are valid, and we get all of Kac’s diagrams. Hope these help, and thanks so much – ガンバテ！ (Japanese “bon courage”)
 * Use Dynkin diagrams, for consistency with the rest of the article/section and to avoid confusion (discussion and diagrams of Coxeter folding would be interesting, but separate) – I think you’re suggesting this above in regards to the Hn.
 * It’s fine if you want to draw the arrows above the edges (instead of on the edge), but it would be clearer if you use double and triple edges rather than numbers.
 * Also, as you suggest, we should eliminate the Hn from the Dynkin article.
 * Use Kac’s notation ($$A_{2l}^{(2)}$$ etc.), since some are from the second series.
 * Add the $$A_{3}^{(1)} \to A_{2}^{(2)}$$ (that’s a square mapping to a quadruple line).
 * I think the “G~2” in E~7 → G~2 ($$G_{2}^{(1)}$$) should instead be $$D_{4}^{(3)},$$ but this is only a guess – it’s the only remaining diagram we don’t otherwise hit, and note that the branch point (valence 3 vertex) in E~7 maps to the center point in the “G~2”, while the branch point in E~6 maps to an outside point.
 * I’m not completely sure how this works, but I think this is right.
 * —Nils von Barth (nbarth) (talk) 10:54, 4 December 2010 (UTC)


 * I'm so glad you're a good careful thinker to can help sort this out. For me, its long past my sleep time. I'd be glad if you want to try your own graphics to replace mine, with corrected notations. Mine can still be helpful as "Coxeter foldings", even if also need some corrections possibly.
 * On E~7--> G~2, I'm convinced it is G~2. Vertically we have a side-ways W zig-zag, which is really a fully folded A5, order 6 symmetry. And the 3 parallel order-3 angles on the right are just 3, so it's [6,3]!

Goodnight. I'm very happy if we can get some sense here! I have emailed some others for advice too.
 * Tom Ruen (talk) 11:23, 4 December 2010 (UTC)


 * p.s. It seems like you could fold A7 into D~4 as a 3-4 zig-zag and merging the 3 nodes into one for a 1-4 branching. Does that make any sense? Can you fold a finite group and get an infinite one?! (Well at least that's similar to my relation of E~7 and E~6 in the G2 folding, seeing the left node column merged or not.) Tom Ruen (talk) 01:10, 5 December 2010 (UTC)


 * The "directed arrow" rule seems to mean to point to the side with less nodes. A perfect zig-zag folding is a folded simplex (An), and will have an odd number of nodes (in Weyl groups), so the arrow is clear. (Hk have 4 nodes, two on each side so no arrow!) The only other cases are like D4-->G2, 3:1, which really is 5 nodes of a 4-simplex foldest as 3:2, with the two nodes merged. So anyway, I think this is clear and my graphs are correct. If I have some time on Sunday evening, maybe I'll try making a chart of Dynkins style graphs (but I don't want to spend too much time on prettiness (like SVG) until its confirmed). Tom Ruen (talk) 02:08, 5 December 2010 (UTC)


 * I was annoyed at an incompleteness, so I took the time to update the W A5 --> G4 folding, and added A3 --> C2/B2 for good measure. Tom Ruen (talk) 04:37, 5 December 2010 (UTC)


 * Okay, I made some Dynkin versions. I kept arrows above the edges, more clear on small scales. So they are ugly, but compact. For me tweaking pixels in MSPaint is less frustrating than SVG. Tom Ruen (talk) 07:01, 5 December 2010 (UTC)

Thanks Tom – the newer versions (red vertical lines, Dynkin versions) are very nice!

I’ll leave graphic design choices up to you – I trust your MSPaint skills, and I well remember all the problems that SVG poses.

The other change is to fix the notation – several of the diagrams (the ones with the “wrong way arrows”) are not extended Dynkin diagrams, but rather other affine Dynkin diagrams. Using the notation on Kac p. 55, the maps should be changed as follows:
 * A~2k–2 → C~k
 * Firstly, the index on this is wrong – it should be $$k-1$$ (remember, extended adds one vertex above the index – this is right in the finite diagram).
 * Secondly, the target is notated by $$D_{k+1}^{(2)}$$ (arrows are opposite of C~).


 * D~k+2 → B~k+1
 * This maps to $$A_{2k+1}^{(2)}$$ – arrow is backwards for B~.


 * E~6 → F~4
 * This maps $$E_6^{(2)}$$ – arrow is backwards for F~4.


 * D~2k+1 → C~k
 * This maps to $$A_{2k}^{(2)}$$ – arrows are >>, not the >< you find in C~.

Two other issues: Thanks again for your continuous improvements!
 * It might also be clearer to replace the ~ by (1), for consistency with the (2)…
 * Also, the bottom right picture in the affine ones isn’t right – you’re taking the quotient of a non-simply laced diagram, which you can’t do (or at least is more delicate) – notice that you’re taking a quotient of a diagram with doubled edges (C~) and the doubled edges disappear in the quotient. Maybe this makes sense, but I’d be very careful.
 * —Nils von Barth (nbarth) (talk) 06:25, 6 December 2010 (UTC)

Thanks! I'm seeing a bit more "Extended" means "~" format by adding one node to finite groups, and the superscripts (1)/(2) define wider directed-graph variations than the original extended definitions. I'm content to use the superscript notations, but FIRST, I think I need an enumeration of these forms before the folding section in the article, like Coxeter-Dynkin diagram summary tables I list all the groups by symbols. Secondly on indexing, extended or not, agreed there's some confusion, but I can't see exactly until things are properly defined. I just did one update, removed one of the lower-right graphs that didn't match in the folding. I have 2-3 extra busy work days this week, so unsure when I can do more. I'll try to more carely make an enumeration table of the groups and names. (Perhaps I should make some parallel Dynkin symbol codes elements like CD, so they can be more easily built up. Less combinations than CD, since no rings!) Tom Ruen (talk) 11:29, 6 December 2010 (UTC)


 * Ah, I found Affine Dynkin diagrams. Good! This superscript syntax is defined at least! I wish I had more time, but at negative time now! Tom Ruen (talk) 11:44, 6 December 2010 (UTC)
 * Cleaned up link formatting. —Nils von Barth (nbarth) (talk) 21:46, 6 December 2010 (UTC)

Agreed, actually listing the affine Dynkin diagrams first (before getting into folding!) makes a lot of sense. No rush – I’m busy myself, and it’s holidays. I won’t close this thread until we’ve finished this, so it shouldn’t slip through the cracks. I’ll see about writing a section on affine Dynkin diagrams (so there’s a good place for the diagrams to go, of course).
 * —Nils von Barth (nbarth) (talk) 21:46, 6 December 2010 (UTC)


 * Hi Nils. I created element symbols for Dynkins, and added summary tables by dimension, but I'm sure there's some arrow problems. I tried to follow what was in the book chart. I see the superscript (2) implies a 2-order folding of a higher node family, but the A/D extensions still don't make sense to me, like the folded ~A family should have arrows pointing in opposite directions. Dynkin_diagram If you can sort them out and correct, I'd be glad. At least the graphs are all enumerated I think, even if the labels (or arrows) are wrongly connected. I'll try to look more at the book on google and see if I can see anything more clearly. Tom Ruen (talk) 20:29, 9 December 2010 (UTC)

Hi Tom – thanks for the fabulous tables!

Agreed, the naming/numbering and arrows for the higher families are really confusing to me too – I’ll see if I can figure out what’s going on.
 * —Nils von Barth (nbarth) (talk) 23:53, 9 December 2010 (UTC)

Arrow confusion
Hi Nils. I've been nervous about the arrow issues, and I found examples of reverse definitions:
 * 1)  Notes on Coxeter transformations and the McKay correspondence, By Rafael Stekolshchik, 2008: Folding examples - Bn arrow directs inwards, Cn, arrow directs outward.
 * 2)  Reflection groups and Coxeter groups, By James E. Humphreys, 1990 - B~n arrow directs outward
 * 3)  Infinite dimensional Lie algebras, Victor G. Kac, 1990 - Bn points outward
 * 4)  Page 7 - finite foldings with arrows and letter names

All but the first seem consistent, and the last defines the arrow directions in relation to foldings, so if I assume that is the correct standard, all my folding arrows are reversed, and B/C relations reversed!

What do you think? Tom Ruen (talk) 04:27, 16 December 2010 (UTC)

I decided to update, corrections seem solid. Tom Ruen (talk) 03:56, 17 December 2010 (UTC)

p.s. Below is a table from the first book above (by Stekolshchik), page 36 (this page not on googlebooks), showing two naming systems, first is in relation to Quiver_(mathematics) usage. I added these also to the test-new chart above left. Tom Ruen (talk) 23:27, 16 December 2010 (UTC)
 * Test_Affine_Dynkin_group_names.png

Dynkin diagrams
Hi Nils! At your convenience, I'm interested in your opinion of a table I added to the talk page: Tom Ruen (talk) 08:20, 9 January 2011 (UTC)
 * Talk:Dynkin_diagram

Euler class, Pontryagin class, Vandermonde determinant, Discriminant
You might be interested in this discussion: Math Overflow. --Kamsa Hapnida (talk) 17:01, 16 October 2014 (UTC)

Levy-Ito decomposition and Lebesgue decomposition
Good afternoon!

You made some changes quite some time ago to Lebesgue decomposition and Lévy_process discussing the relation between them. Do you have any references for the statements that you added? I have left quite a detailed comment at Talk:Lévy_process explaining why I don't think the relation is precise as stated - perhaps it is an interesting analogy, but I wasn't able to see a way to make it into a one-to-one correspondence. What do you think? Thanks! Alex. 130.88.123.107 (talk) 17:26, 18 January 2017 (UTC)

Sorry about that! You're right, it's just an analogy. I'll make that clearer, and elaborate that they have different continuity properties, with citations, as you indicate. I'm currently traveling, so feel free to go ahead with changes, but I'll clean up and notify you for review. Thanks again! —Nils von Barth (nbarth) (talk) 16:37, 15 February 2017 (UTC)

PDIFF as a category
Hi — in 2009 you added some material regarding PDIFF to piecewise linear manifold. Today I received an email from one of my coauthors complaining that although one can talk about piecewise smooth manifolds on an individual basis, PDIFF is not really a category, or at least the sources do not support it being a category. I have to admit that to me the sourcing to PDIFF looks weak. Do you know more about this? Is there justification for calling it a category despite Google scholar finding almost nothing when one searches for "category of piecewise smooth manifolds" (or differentiable in place of smooth)? If not, should the category-theoretic language surrounding this material be removed? —David Eppstein (talk) 05:00, 1 June 2018 (UTC)


 * Hi David,
 * I'm traveling now, so I can get back better in a week or so.
 * The main reason for the lack of references is that it's a minor technical point, hence no one bothers working it out; it's oral folklore.
 * There are probably a few subtleties in making it precise; Thurston (p. 194) writes:
 * Show that [maps from Rn to itself] don't form a group. Thus, piecewise smooth maps serve as bridges between piecewise linear and smooth structures; they don't work well alone.
 * That said, he calls the self piecewise smooth homeomorphisms a pseudogroup, so I think the category axioms (identity obviously, and composition) are fulfilled, but inverting doesn't work (unlike in PL)?
 * I assume the only objection to this being a category is that it's not obvious that a composition of piecewise smooth maps is piecewise smooth, because the triangulations don't agree (gives maps from X to Y to Z, the triangulation on Y may not agree with the image of the triangulation on X, or rather not pull back properly). I think this is the same issue for PL, and the answer is to actually have equivalence classes of piecewise linear/smooth structures.
 * Basically all the "essentially unique" results predate the popularization of category theory, hence they're not stated in that language.
 * I haven't been working in math since grad school (over 10 years ago), so I'm not the best person to ask.
 * Probably best would be to ask Curt (Curtis T. McMullen, ctm@math.harvard.edu), whom I'm referencing (Re: PL and DIFF manifolds: a question), or my advisor Shmuel Weinberger (shmuel@math.uchicago.edu); if you mention me they'll recognize it, and might be able to clarify. (They're both quite prominent, and if there's a mistake or gap in the literature, Curt might be inclined to rectify it.)
 * I can have a shot at rewording it so it's more accurate about what the literature actually states.
 * —Nils von Barth (nbarth) (talk) 03:18, 2 June 2018 (UTC)
 * It's the composition my correspondent is worried about. The problem is that unless there's more to the definition than the obvious, the triangulations may not agree, badly enough that you can't piece them together into a common refinement. —David Eppstein (talk) 04:36, 2 June 2018 (UTC)

Permutation representation
Bonjour, j'ai vu que plusieurs liens vers la page de redirection Permutation representation (symmetric group) (vers la théorie des représentation linéaires du groupe symétrique) faisaient en fait référence à permutation representation. J'ai corrigé ces erreurs mais il me semble que cette page est un peu hors-propos (je n'ai jamais vu cet usage pour "permutation representation---mais je ne suis pas un spécialiste en groupes finis), et que le mieux serait de la supprimer (en corrigeant les quelques liens vers qui restent). Cordialement, jraimbau (talk) 19:18, 21 May 2019 (UTC)

Massey product
Dear Nils,

I saw that you added in 2009 a geometric picture about the Massey triple product. I am very interested in learning more about this geometric interpretation. I have searched a lot about the topic, but I have only found the geometrical interpretation of cohomology products via Borel-Moore cohomology. Do you have a reference about the geometrical interpretation of Massey triple products?.

Thanks in advance. — Preceding unsigned comment added by Jlleon (talk • contribs) 21:32, 9 February 2021 (UTC)

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