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Is there a "canonical" set of generators for a given coxeter group? If so, is there a method for going from an arbitrary set of generators of the group to the canonical?

(The "textbook" definition doesn't include this in the definition of these groups, although it certainly seems to use them.)


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The notion of "Coxeter group" incorporates a particular choice of generators and relations for the group. Examples show that the underlying abstract group may well be given by two distinct sets of Coxeter data, which in turn complicates the question of studying isomorphisms and automorphisms. So I'm not sure what the question here actually means. Keep in mind that even a finite symmetric group can usually be given by quite different sets of generators and relations, so a Coxeter presentation of that abstract group is rather special. –  Jim Humphreys Feb 26 '11 at 21:46

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up vote 11 down vote accepted

There are isomorphic Coxeter groups with different Coxeter diagrams. So a simple answer to your question is "no". Nevertheless, the sets of isomorphism classes of Coxeter groups given by Coxeter diagrams are not very large, and that information can be viewed as the "almost yes" answer to your question. See, for example Mihalik, Michael, Ratcliffe, John, Tschantz, Steven, Quotient isomorphism invariants of a finitely generated Coxeter group. Aspects of infinite groups, 212–227, Algebra Discrete Math., 1, World Sci. Publ., Hackensack, NJ, 2008 or Marquis, Timothée, Mühlherr, Bernhard, Angle-deformations in Coxeter groups. Algebr. Geom. Topol. 8 (2008), no. 4, 2175–2208 and the references there.

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These papers and related recent ones are available on the arXiv, along with a useful older survey paper on the isomorphism problem by Muhlherr: front.math.ucdavis.edu/0506.5572 –  Jim Humphreys Feb 27 '11 at 14:30
Thank you. I am going through these to see if I can find the answer to what I am looking for. –  P.H. Mar 1 '11 at 19:20

My answer is definitely less complete than that of Mark Sapir, but in case you want to see an explicit example: the easiest counterexample is the dihedral group $D_{12}$, which you can view either as a Coxeter group of type $G_2$ (with $2$ generators), or as a Coxeter group of type $A_1 \times A_2$ (with $3$ generators).

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