Talk:Coxeter group/Archive 1

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Archive 1

old stuff that needs a heading

Possible improvements:

Describe dihedral groups explicitly as finite Coxeter groups in the Euclidean plane, and explain the geometric meaning of the relator.
Describe the way in which the symmetric group $S_{n}$ acts by reflections in Euclidean n-space.
State the theorem on special subgroups: each subset of the generators generates a subgroup isomorphic to a Coxeter group.
Say the exact manner in which the group acts by affine reflections: the "geometric representation".
Give the exact list of which Coxeter groups act discretely by isometric reflections in Euclidean space of some dimension. This list is not too much more complicated than the list of finite Coxeter groups.
Explain how some Coxeter groups (such as certain triangle groups) act discretely in hyperbolic space of some dimension, generated by isometric reflections.

--Mosher 10:48, 8 October 2005 (UTC)

Rank

The rank of a Coxeter group is not defined. —Preceding unsigned comment added by 81.210.238.84 (talk) 08:03, 15 April 2008 (UTC)

Update notice

Does anyone watching have an interest in this page? I've expanded an article Coxeter-Dynkin diagram, using Coxeter groups, but a different letters than shown here - used in his book Regular polytopes. Looks like diagrams here are the original letters which Coxeter renamed a bit to A-G for finite groups, and P-W for infinite groups.

I'd like to update this article to reflect this different system. I'm happy to make a conversion table showing the old/new systems. I'd like all the uniform polytope articles to reference Coxeter group and the group name.

I'll hold for a week for responses. Thanks! Tom Ruen 04:50, 25 January 2007 (UTC)

Please do not make those changes here. The terminology on the other page has some serious mistakes and does not align itself with how most of the mathematics community views Dynkin diagrams and Coxeter graphs. I will address my problems with the other page when I have sufficient time. (unkown editor)

Indeed, I think it's a big mistake to use this "alternate" convention with "P-W" letters, even if Coxeter used those in his original work. All modern books and papers dealing with Coxeter groups and diagrams that I am aware of (and I am aware of quite a few, working on building geometries) uses the notation corresponding to that used for finite simple groups. There are good systematic reasons for writing e.g. A~_n instead of P_n, too (because the Coxeter group associated to the A~_n diagram has A_n residues, for example). Since virtually everything in the literature uses the "new" system (and has been doing so for several decades), it will only confuse users to read the "alternate" names. In fact, I'd suggest to omit any mention of the alternate names (or at least delegate them to a "Historical notes" section). BlackFingolfin (talk) 12:34, 16 August 2008 (UTC)

mislabel?

Okay, I've just changed the table in the Affine Weyl Groups section. This is because in the Humphreys text which is referenced at the bottom of the article, the labels B~_n and C~_n are the other way round from how they had been here. They've already been swapped in the Coxeter-Dynkin diagram article where I brought this up. I don't know much about PNG files though, so for the time being the figure in the Affine Weyl Groups section disagrees with the table next to it.137.205.31.198 (talk) 13:47, 16 December 2008 (UTC)

I changed the PNG - swapped the B/C labels. Tom Ruen (talk) 16:38, 16 December 2008 (UTC)

More explanation needed

There are several things about this page that seem incomplete, or wrong. I don't know enough to be able to fix them myself, so I am hoping someone else can.

The presentation of a Coxeter group is given as $\left\langle r_{1},r_{2},\ldots ,r_{n}\mid (r_{i}r_{j})^{m_{ij}}=1\right\rangle$ . Should this be $\left\langle r_{1},r_{2},\ldots ,r_{n}\mid r_{i}^{2}=1,(r_{i}r_{j})^{m_{ij}}=1\right\rangle$ ?
The examples use what appear to be down-arrows in Coxeter-Dynkin diagrams. The article doesn't explain what these mean. In fact I think they are just ordinary nodes on the end of side-branches, but they look to me squarer than the other nodes. It would help if they were explained.
The Petrie polygon article uses ringed nodes like this in Coxeter diagrams, with no explanation of their meaning. It should be explained either in this article or there.
The section "An example" states "The graph in which vertices 1 through n are placed in a row with each vertex connected by an unlabelled edge to its immediate neighbors gives rise to the symmetric group S_n+1". But the section "Finite Coxeter groups - Classification" starts with a diagram showing ten Coxeter graphs. The first of these is n vertices linked in a row as described, and is labelled "A_n". I assume, despite the roman A, that this is a Chevalley group not an alternating group; but either way, A_n or A_n isn't the same as S_n+1.

Maproom (talk) 12:17, 9 February 2010 (UTC)

Here are some answers:

Both presentations are the same group. The second presentation lists some of the relators twice. Since m_ii=1, (r_ir_i)¹ = r_i² = 1.
For coxeter groups in group theory, all the nodes are the same. The ringed stuff is for geometry.
(no comment, but I recall it is explained on one of the "Coxeter" pages in wikipedia)
The coxeter group An is the symmetric group on n+1 points. It is the Weyl group of the groups PSL(n+1,?), SL(n+1,?), PGL(n+1,?), etc. which are Lie/Chevalley/algebraic groups of type An. The alternating group is the "rotation subgroup" of the coxeter group of type An, but this is only useful in certain contexts. Obviously some letters are very popular names and lots of notation conflicts. The dihedral groups of order 2n are Coxeter groups of type I2(n), and the Coxeter group of type Dn is an index 2 subgroup of the hyperoctahedral group, the Coxeter group of type Bn.

Hope this helps, JackSchmidt (talk) 15:09, 9 February 2010 (UTC)

Thank you, that helps a lot.

Point 1: yes, I should have realised that.

Your point 2 my point 3: ok, understood.

Your point 3 my point 2: maybe I could go to commons and use an SVG program to redraw

and

to make the latter more blobby and less arrow-like. There's only three pages that use them, I would check that they still looked right on those.

Point 4: I understand. As a check I have now compared the diagram in the article with page 72 of "Abstract regular polytopes" by McMullen & Schulte. I find there are differences:

A_n here corresponds to A_n in ARM.
BC_n here corresponds to C_n in ARM.
D_n here corresponds to B_n in ARM.
E₆, E₇, E₈, F₄, H₃, H₄ all match up right.
I_n here corresponds to D₂ⁿ in ARM.

Does this matter? I could redraw the diagram here to show what I think is the more usual use of these labels. I'll take your advice. Maproom (talk) 16:29, 9 February 2010 (UTC)

JackSchmidt writes above:

"Both presentations are the same group. The second presentation lists some of the relators twice. Since m_ii=1, (r_i r_i)1 = r_i^2 = 1."

I feel the definition in the article needs some work. The problem is with the presence or absence of quantifiers. The relation (r_ir_j)^m_ij = 1 is not quantified by all i and all j (nor should it be) -- at least not the way it is interpreted in a standard presentation.

(By the way, in a standard presentation, the right hand side need only contain the "relators" -- those words in the generators that are assumed to be set equal to the identity; the full equation -- that they = 1 -- does not need to be mentioned, and in fact should not be.)

But the presentation in the article assumes without saying so that all m_ii are covered by the statement m_ii = 1.

A further problem with the presentation given in the article is that it is nowhere mentioned that m_ij = m_ji for all i and j. This is part of the standard definition of a Coxeter group (as found in, e.g., Springer Online Reference, or the book Abstract Regular Polytopes).

The wording of the article leaves these things less than clear. I volunteer to edit them to improve accuracy and clarity, but I won't be able to get around to it for a couple of weeks.Daqu (talk) 18:13, 12 March 2010 (UTC)