Talk:Coxeter group

message: The article defines Coxeter group as follows:
"Formally, a Coxeter group can be defined as a group with the presentation
〈 r_1, r_2, . . ., r_n  |  (r_i r_j)^m_ij 〉
where m_ii = 1 and  m_ij >= 2 for i ≠ j. The condition m_ij = ∞ means no relation of the form (r_i r_j)^m should be imposed."
The exact meaning of this definition can probably be guessed from reading further down the page.  But why should anyone have to do that when a few more statements would make this definition clear, instead of unclear???  
For one thing, no quantifier is applied to the subscripts of m_ij in the presentation, nor to the subscripts of m_ii in the next clause: Why not?  And why not state *somewhere* that m_ij belongs to ℤ ∪ {∞} ???  
Also: the last statement is mysterious, since it is unclear how "the condition m_ij = ∞" can be part of the presentation of the group (and there is nowhere else for this condition to be imposed). Probably this is backwards: what is intended is likely "If no relation of the form (r_i r_j)^m_ij appears in the presentation, then the number m_ij is taken to be ∞."  If so, this should be stated clearly.Daqu (talk) 16:41, 29 January 2010 (UTC)
replies: text: There are many groups which are generated by involutions and which are not Coxeter groups. For example, (almost?) all non-abelian finite simple groups are generated by involutions. So your suggestion does not work. One really needs to define Coxeter groups using the m_ij. BlackFingolfin (talk) 20:44, 11 January 2011 (UTC)
thread_title:  Let's define "Coxeter group" properly 

message: Is there a reason why this page used m_{ij} = \infty to mean "there is no relation between r_i and r_j. Surely it would be more logical and less scary to write m_{ij} = 0 (the relation (r_ir_j)^{m_{ij}} = 1 would just reduce to (r_ir_j)^0 = 1 = 1), or have I missed something? 129.11.105.213 (talk) 16:23, 28 February 2010 (UTC)
replies: text: I guess m_{ij} = 0 is technically correct, but it doesn't sound helpful. The question is - what is the order of the group element (r_ir_j)? or, how many times do we have to multiply the identity by (r_ir_j) to get back to the identity?  In this case we will never get back to the identity, so "infinity" seems to me a more helpful answer than "0". Maproom (talk) 20:38, 28 February 2010 (UTC)
replies: text: No, the exponent 0 does not add any additional information. The reason mij is assigned the "value" ∞ (for those products rirj that do not appear among the relators in the presentation) is solely for convenience.  For when an element g of a group G has no power gn that is equal to the identity 1G (for any positive integer n), then as a shorthand, g is said to have "order infinity".  This should not be interpreted to mean that "g∞ = 1G". (Which would be considered a meaningless expression.)Daqu (talk) 18:26, 12 March 2010 (UTC)
thread_title:  Why Infinity? 

message: Any one care to explain the reasoning for the naming scheme of the graphs? Jka02 (talk) 15:26, 28 March 2012 (UTC)
thread_title: Naming scheme of Coxeter graphs diagrams

message: If it's known, it might be nice to put up the relationship between the graph and the corresponding classification by finite simple groups? If not, I think GAP might be able to do this?  Jka02 (talk) 15:26, 28 March 2012 (UTC)
thread_title: Relationship between Coxeter group and diagram

message: At some point in time a little discussion about the connection between the Cartan matrix and the bilinear form would be worth adding. Jka02 (talk) 15:26, 28 March 2012 (UTC)
thread_title: Connection between bilinear form and Cartan matrix

message: We are told all finitely generated Coxeter groups are automatic. Sure, but why specify finitely generated - that's in the definition as given. Is this not assuming a more general convention?  — Preceding unsigned comment added by 75.38.193.168 (talk) 01:55, 4 November 2011 (UTC)
thread_title:  Finitely generated? 

message: In the example in the table the Cartan matrix for B3 has sqrt(2) in it!!!
--surely it should be non symmetric and have integral entries?  — Preceding unsigned comment added by 158.109.1.18 (talk) 17:27, 29 May 2012 (UTC)
replies: text: Coxeter groups are undirected, so the Cartan entries are symmetric. Tom Ruen (talk) 19:53, 29 May 2012 (UTC)
thread_title:  B3 cartan matrix?? 

message: In the table of Coxeter matrices and Cartan Matrices, and nowhere else in the article, the Cartan matrices are called "Schlaefli" matrices. Ericlord (talk) 07:10, 19 February 2013 (UTC)
replies: text: I changed it to Schläfli matrix which redirects to a section at Coxeter-Dynkin diagram. Tom Ruen (talk) 20:06, 19 February 2013 (UTC)
thread_title:  Cartan Matrices 

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thread_title:  External links modified 

message: Table has:
Script error: No such module "CDD".
No further details are available.  — Preceding unsigned comment added by 75.4.201.47 (talk) 03:39, 27 February 2019 (UTC)
replies: text: Caused by a name change, seems to fix on refresh. Tom Ruen (talk) 10:06, 27 February 2019 (UTC)
thread_title:  Bad Links