Timeline for Embed one Coxeter System into another
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 15, 2014 at 23:45 | comment | added | Geordie Williamson | @Jim: I would be very interested in a reference earlier than the Lusztig ref (as would Lusztig I guess). I think it is clear that Lusztig is aware of the general pattern that such embeddings follow. I agree the history is murky! Also +1 for "misleadingly named"! | |
| Aug 15, 2014 at 22:59 | comment | added | Jim Humphreys | @Geordie: The history looks murky to me, but I have been revisiting the past literature. For example, the 1988 article by Shcherbak (translated in Russian Math Surveys) has a more complete series of diagrams including Lusztig's. But too little history is given there. I never looked deeply into these things 25 years ago, lacking motivation. Bourbaki apparently goes only as far as Witt's construction of $H_3, H_4$ using quaternions. In any case, Lusztig's example is just one of the many fascinating things in that (misleadingly named) paper. | |
| Aug 15, 2014 at 21:20 | comment | added | Sam Hopkins | @DavidSpeyer: I think that manuscript of Stembridge only addresses embeddings of crystollagraphic root systems in simply-laced ones, in which case there is an easy way to see the embedding via "folding" which arises from the fixed-point set of an automorphism of the Coxeter graph. But the embeddings discussed by the OP (of $H_3$ into $D_6$ or $H_4$ into $E_8$) are different and more complicated than this. | |
| Aug 15, 2014 at 21:00 | comment | added | David E Speyer | Duplicate of mathoverflow.net/questions/138083 ? Whether or not it is a duplicate, Stembridge's manuscript math.lsa.umich.edu/~jrs/papers/folding.ps.gz seems like a good answer to both questions. | |
| Aug 15, 2014 at 14:03 | comment | added | Jim Humphreys | @Geordie: Lusztig certainly gives an insightful embedding of $H_4$ into $E_8$ (though not the first such). Here is an online link to the article: ams.org/journals/tran/1983-277-02/S0002-9947-1983-0694380-4… | |
| Aug 15, 2014 at 12:52 | comment | added | Nathan Reading | @Geordie: You should "promote" this comment to an answer. I suspect this is exactly what Qiao wants. | |
| Aug 15, 2014 at 7:36 | comment | added | Geordie Williamson | If I remember correctly these embeddings are discussed (for the first time?) in Lusztig, "Some examples of square integrable representations of a p-adic group". See discussion around p. 636 and lovely diagram... | |
| Aug 14, 2014 at 19:08 | answer | added | Christian Stump | timeline score: 0 | |
| Aug 14, 2014 at 19:07 | answer | added | Jim Humphreys | timeline score: 6 | |
| Aug 14, 2014 at 16:29 | history | asked | Qiao | CC BY-SA 3.0 |