Timeline for Coxeter subgroups of Coxeter groups
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Sep 26, 2021 at 9:58 | answer | added | AGenevois | timeline score: 3 | |
| Jun 11, 2021 at 2:37 | comment | added | David E Speyer | Not that I know a method of classifying finite index reflection subgroups of Coxeter groups, but it seems like it might be a problem of manageable size. | |
| Jun 11, 2021 at 2:36 | comment | added | David E Speyer | I wonder whether you might be looking for the notion of a reflection subgroup? Recall that an element of a Coxeter group $W$ is called a reflection if it is conjugate to a simple generator; a theorem of Deodhar mathscinet.ams.org/mathscinet-getitem?mr=1023969 and Dyer mathscinet.ams.org/mathscinet-getitem?mr=1076077 states that a subgroup of a Coxeter group generated by reflections will always be a Coxeter group. | |
| Jun 11, 2021 at 0:11 | comment | added | ArB | Do Coxeter groups tend to have a lot of normal subgroups? | |
| Jun 10, 2021 at 23:07 | comment | added | paul garrett | Just echoing other comments: an algorithm or semi-algorithm to determine all Coxeter groups inside some $S_n$ is surely implausible. But perhaps this is not the real question? "All subgroups which are Coxeter groups and [something]" might be answerable in a useful way... Comment? | |
| Jun 10, 2021 at 21:08 | comment | added | Stefan Witzel | Is this a question about abstract or about practical computability? It seems abstractly this is essentially about deciding whether a given finitely presented group is Coxeter, isn't it? Practically, thinking just of S_n is disheartening. | |
| Jun 10, 2021 at 21:05 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title
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| Feb 11, 2021 at 2:26 | answer | added | Daniel Sebald | timeline score: -1 | |
| Aug 16, 2015 at 10:17 | comment | added | YCor | Related question: given two finite Coxeter graphs, determine whether the associated Coxeter groups are abstractly commensurable (that is, admit isomorphic subgroup of finite index) | |
| Aug 14, 2015 at 8:40 | comment | added | Derek Holt | But if the given Coxeter group is finite then the answer to the question is clearly yes. Also the subgroups of finite index that are isomorphic to a Coxeter group is recursively enumerable. I would guess that the answer to the general questionis no. | |
| S Aug 13, 2015 at 20:09 | history | suggested | Tadashi |
Added relevant tag
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| Aug 13, 2015 at 19:49 | review | Suggested edits | |||
| S Aug 13, 2015 at 20:09 | |||||
| Aug 13, 2015 at 19:01 | comment | added | i. m. soloveichik | By Coxeter subgroup I mean that you specify involution generators with presentation given by the relations of a Coxeter diagram. | |
| Aug 13, 2015 at 18:56 | comment | added | Jim Humphreys | Be careful about the formulation: being a "Coxeter group" requires fixing a set of involutive generators. However, a finite symmetric group $S_n$ will typically contain a lot of smaller Coxeter groups whose generators have nothing to do with those of $S_n$ itself, since every finite group has some embedding in a symmetric group. In another direction, a subgroup of a Coxeter group generated by a finite set of "reflections" (conjugates of the given generators) will be a Coxeter group relative to this new set of involutions (Deodhar, Dyer). Many possibilities. | |
| Aug 13, 2015 at 18:19 | history | asked | i. m. soloveichik | CC BY-SA 3.0 |