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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.)

Thanks.

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    $\begingroup$ 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. $\endgroup$
    – Jim Humphreys
    Commented Feb 26, 2011 at 21:46
  • $\begingroup$ It depends on your meaning of "given Coxeter group" (and maybe of "canonical"). If you start from a group which you only know to be isomorphic to a Coxeter group, the only "canonical" generating subsets should be invariant under automorphism (and hence under conjugation); in practice such generating subsets are very large compared to useful ones. $\endgroup$
    – YCor
    Commented May 5, 2020 at 6:13

2 Answers 2

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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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    $\begingroup$ 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 $\endgroup$
    – Jim Humphreys
    Commented Feb 27, 2011 at 14:30
  • $\begingroup$ Thank you. I am going through these to see if I can find the answer to what I am looking for. $\endgroup$
    – P.H.
    Commented Mar 1, 2011 at 19:20
  • $\begingroup$ The link in Jim's comment is broken, here's a replacement: arxiv.org/abs/math/0506572 $\endgroup$
    – David Roberts ♦
    Commented Mar 29, 2022 at 7:09
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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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  • $\begingroup$ Is there an easy example where the "Coxeter ranks" of the two Coxeter presentations are the same? $\endgroup$
    – LSpice
    Commented Aug 3, 2020 at 22:54
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    $\begingroup$ Not as easy as the above example, but the one-page paper "On Isomorphisms between Coxeter Groups" by Bernhard Mühlherr (doi.org/10.1023/A:1008347930052) gives an explicit example of two non-isomorphic irreducible Coxeter systems of rank 4 for which the resulting groups are isomorphic. $\endgroup$
    – Tom De Medts
    Commented Aug 4, 2020 at 13:23

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