Timeline for Is there any need to study Coxeter systems (W,S) with S infinite?
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Aug 21, 2013 at 18:33 | answer | added | Ben | timeline score: 7 | |
| Aug 20, 2013 at 20:17 | answer | added | David E Speyer | timeline score: 12 | |
| Feb 3, 2013 at 14:51 | comment | added | Jim Humphreys | @Tom: Thanks for reminding me about Nuida's work. But I'm not quite ready to apply my elusive adjective "significant" to the results there on infinite rank Coxeter groups. Coming from the direction of Lie theory and its deep results depending on finite rank Coxeter groups, I have some hesitation about what is important in Nuida's work (as well as the work outlined by Yves, which however strikes me initially as more substantial). | |
| Feb 1, 2013 at 9:02 | comment | added | Tom De Medts | I'm not sure whether it answers your question, but Koji Nuida has done quite a bit of research about infinite rank Coxeter groups: www2u.biglobe.ne.jp/~nuida/m/works_e.htm , in particular in his PhD thesis www2u.biglobe.ne.jp/~nuida/m/thesis_text.dvi . | |
| Feb 1, 2013 at 8:43 | comment | added | YCor | @Benjamin OK I see. | |
| Feb 1, 2013 at 4:21 | comment | added | Benjamin Steinberg | @yves, my point is that infinite rank Coxeter groups are direct limits of finite rank Coxeter groups and so combinatorial statements involving finitely many elements can be deduced from the finite rank case. I didn't mean to imply they are locally finite. | |
| Feb 1, 2013 at 0:13 | vote | accept | Jim Humphreys | ||
| Jan 31, 2013 at 23:35 | comment | added | YCor | @Benjamin: I don't agree: the infinite finitary symmetric group seems to be quite exceptional among infinitely generated Coxeter groups, since it has the property that all finite rank parabolics are finite. There are only 4 irreducible infinite rank Coxeter systems with this property namely: the two $A_\infty$ (one-sided infinite or two-sided infinite), $B_\infty$, and $D_\infty$. Any other irreducible infinite rank Coxeter group contains a free subgroup on 2 generators! | |
| Jan 31, 2013 at 23:29 | comment | added | YCor | Note that a search on "infinitely generated Coxeter" provides many more answers than "infinite rank Coxeter". | |
| Jan 31, 2013 at 14:22 | comment | added | Lee Mosher | @Jim: On the "hard sell" issue, as a pedagogical technique I would not hide the fact that it all works in arbitrary cardinality, as long as it is true that finiteness of $S$ is never actually used, even if I wouldn't bother taking the time in a course or in advisement to talk about specific infinitely generated examples. | |
| Jan 31, 2013 at 13:55 | comment | added | Jim Humphreys | @pranavk: This confirms my suspicions. (I never thought to ask any of the insiders decades ago.) In mathematics one often looks for the natural generality in which to state things, but that can be a hard sell to students who need to get on to their own work. | |
| Jan 31, 2013 at 13:52 | comment | added | Jim Humphreys | @Misha: Certainly the infinite rank groups do appear "naturally", but as you point out there isn't much Coxeter theory used. My word "significant" is of course not defined mathematically, but I was motivated especially by the restriction to finite rank in work on Soergel bimodules and in overlapping work on "realizations" of Coxeter groups by matrices. | |
| Jan 31, 2013 at 12:58 | answer | added | YCor | timeline score: 8 | |
| Jan 31, 2013 at 10:43 | comment | added | Benjamin Steinberg | Isn't an infinite rank Coxeter group just the direct limit of its finite rank parabolics and so the situation is always like the infinite symmetric group? | |
| Jan 31, 2013 at 5:27 | comment | added | user30379 | I once asked a member of the 2nd generation of Bourbaki why they decided to develop the theory without assuming $S$ to be finite. The answer I got was that the proofs work for general $S$, so they decided to write it that way (and that there were no specific examples of interest with infinite $S$ that they had in mind). | |
| Jan 31, 2013 at 5:06 | comment | added | Misha | Infinite rank (even of cardinality continuum) Coxeter groups appear naturally in the theory of nondiscrete affine buildings, e.g. asymptotic cones if symmetric spaces. However, the Coxeter theory in this setting is rather trivial. | |
| Jan 31, 2013 at 0:05 | history | asked | Jim Humphreys | CC BY-SA 3.0 |