# Talk:Dynkin diagram

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Field:  Algebra

## Changed redirect

The original link was to root systems. I have changed it to Coxeter–Dynkin diagram. Although they do use Coxeter–Dynkin diagrams to classify root system, a root system is not a Coxeter–Dynkin diagram. Dharma6662000 (talk) 19:18, 20 August 2008 (UTC)

Agreed – I’d even go further and suggest that Dynkin diagrams merit their own page, as they are distinct from but confusingly similar to Coxeter diagrams of finite groups. I’ve accordingly marked the redirect as “with possibilities”. (And significantly expanded the discussion of Dynkin diagrams.)
—Nils von Barth (nbarth) (talk) 08:01, 26 November 2009 (UTC)
I’ve now done so (moved it to its own page); could use some work, but now it’s no longer threatening to overwhelm the Coxeter group page.
—Nils von Barth (nbarth) (talk) 07:14, 1 December 2009 (UTC)

## Meaning of branches

I'm still verifying this, but added the Cartan matrix and (un)folded simply-connected graph equivalent, These seem to represent a near perfect correspondence to the Dynkins branching notation, AND the Cartan matrix nondiagonal elements seem to correspond to the number of nodes in the folding as well. I don't know if anyone has tried to present it like this before, but it seems very helpful! Tom Ruen (talk) 08:23, 9 January 2011 (UTC)

I added a column "value graph", apparently primarily used for hyperbolic graphs. It is defined and used in Notes on Coxeter Transformations and the McKay correspondence, Rafael Stekolshchik, 2005 [1]. Tom Ruen (talk) 21:44, 10 January 2011 (UTC)
I moved the summary table to a new article section, Dynkin diagram#Rank 2 Dynkin_diagrams. More integration is needed in the article, and perhaps it should be moved down, or duplicated into a smaller introductory table? Tom Ruen (talk) 03:23, 11 January 2011 (UTC)
Rank 2 Dynkin diagrams
Group
name
Dynkin diagram Cartan matrix Symmetry
order
Related
simply-laced
automorphic
group3
(Standard)
multi-edged
graph1
Valued
graph2
${\displaystyle \left[{\begin{matrix}2&a_{12}\\a_{21}&2\end{matrix}}\right]}$ Determinant

(4-a21*a12)

Finite (Determinant>0)
A1xA1 ${\displaystyle \left[{\begin{smallmatrix}2&0\\0&2\end{smallmatrix}}\right]}$ 4 2
A2 ${\displaystyle \left[{\begin{smallmatrix}2&-1\\-1&2\end{smallmatrix}}\right]}$ 3 3
B2 ${\displaystyle \left[{\begin{smallmatrix}2&-2\\-1&2\end{smallmatrix}}\right]}$ 2 4 ${\displaystyle {A}_{3}}$
C2 ${\displaystyle \left[{\begin{smallmatrix}2&-1\\-2&2\end{smallmatrix}}\right]}$ 2 4 ${\displaystyle {A}_{3}}$
G2 ${\displaystyle \left[{\begin{smallmatrix}2&-1\\-3&2\end{smallmatrix}}\right]}$ 1 6 ${\displaystyle {D}_{4}}$
Affine (Determinant=0)
A1(1) ${\displaystyle \left[{\begin{smallmatrix}2&-2\\-2&2\end{smallmatrix}}\right]}$ 0 ${\displaystyle {\tilde {A}}_{3}}$
A2(2) ${\displaystyle \left[{\begin{smallmatrix}2&-1\\-4&2\end{smallmatrix}}\right]}$ 0 ${\displaystyle {\tilde {D}}_{4}}$
Hyperbolic (Determinant<0)
${\displaystyle \left[{\begin{smallmatrix}2&-1\\-5&2\end{smallmatrix}}\right]}$ -1 H5(6)
${\displaystyle \left[{\begin{smallmatrix}2&-b\\-a&2\end{smallmatrix}}\right]}$ 4-ab

Note1: The multi-edged diagram correponds to the nondiagonal Cartan matrix elements a21, a12, with the number of edges drawn equal to max(a21, a12), and an arrow pointing towards nonunity element(s).

Note2: For hyperbolic groups, (a12*a21>4), the multiedge style is abandoned in favor of an explicit labeling (a21, a12) on the edge. These are usually not applied to finite and affine graphs.

Note3: Many multi-edged groups are automorphic via a folding operation with a higher ranked simply-laced group.

## Computation of folding

It looks to me that geometric folding can be defined by multiplication of a Cartan matrix by a mapping function. For example this shows the mapping of the D4 group into G2. The 4x2 mapping matrix defines which nodes map into the same lower dimensional node. The last step removes the 3rd and 4th rows which are the same as the first row. If someone has some documentation for this process it would be nice to write up in the article section on folding. Tom Ruen (talk) 04:11, 16 January 2011 (UTC)

D4 Fold
mapping
3,4-->1
D4 G2
${\displaystyle \left[{\begin{matrix}2&-1&0&0\\-1&2&-1&-1\\0&-1&2&0\\0&-1&0&2\end{matrix}}\right]}$ x ${\displaystyle \left[{\begin{matrix}1&0\\0&1\\1&0\\1&0\end{matrix}}\right]}$ = ${\displaystyle \left[{\begin{matrix}2&-1\\-3&2\\2&-1\\2&-1\end{matrix}}\right]}$ --> ${\displaystyle \left[{\begin{matrix}2&-1\\-3&2\end{matrix}}\right]}$

## Arrow direction

There is a severe unclearity concerning the decoration of the Dynkin diagram arrow direction: While the top of the page shows the arrow pointing to the shorter root (e.g. in B_n "outwards" to the unique short root), the diagrams later have an arrow on top of the line pointing to the longer root (e.g. "inwards" for B_n), and the similarly-looking diagrams below on this discussion page again have the arrow pointing to the short root according to the Cartan matrix (e.g. B_2 left-to-right), while the folding in the last column is again flipped (the split node becomes longer, i.e. should stand left in B_2). The latter seems to be a "consequence-error" from using the former diagrams. Please crosscheck this observation and adapt the pictures accordingly! (resigned with newly created account) Pacman 2.0 (talk)

Thanks for the note. There was an arrow confusion from different sources, so all should have been corrected to the standard. I changed in the folding section and associated diagram. Tom Ruen (talk) 23:29, 30 May 2011 (UTC)

## Twisted affine type A diagrams

I don't think the indexing of the twisted affine type A diagrams (i.e., A(2)2k and A(2)2k-1) are correct in the figure.

A(2)2k-1 is dual to B(1)k, which is the diagram now labelled by A(2)2k. Also note that this diagram will have k+1 vertices (or, k yellow vertices) contrary to what the text below the diagram says (that it will have k vertices).

A(2)2k then corresponds to the other diagram. I think it also corresponds to a diagram with k yellow vertices (k+1 in total) but I don't have a good reference for that.

Ctourneur (talk) 18:44, 19 May 2012 (UTC)

Found the reference: Kac, Infinite dimensional Lie algebras, the table on page 55. (Previewable on Google books.) It confirms what I said.

Ctourneur (talk) 18:58, 19 May 2012 (UTC)

The Kac reference also makes clear that D(2)k+1 also gives a diagram with k yellow nodes (k+1 in total).

I am going to go ahead and make the change from "k nodes" to "k yellow nodes"; since the other change requires changing the diagram, which I can't easily do, I'm going to hold off on that.

Ctourneur (talk) 19:23, 19 May 2012 (UTC)