Timeline for Reference for embeddings of reflection groups (related to folding ADE Coxeter graphs)?
Current License: CC BY-SA 4.0
17 events
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| Jan 8, 2021 at 17:27 | history | edited | YCor | CC BY-SA 4.0 |
fixed typo (the question was just bumped)
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| Jan 8, 2021 at 16:41 | answer | added | Communicative Algebra | timeline score: 3 | |
| Jan 5, 2021 at 12:05 | comment | added | Communicative Algebra | @NathanReading Stembridge assumes that simple roots in the same orbit are orthogonal. This is not the case for the non-trivial automorphism of $A_2$, $A_4$, … | |
| Nov 22, 2013 at 16:36 | comment | added | Nathan Reading | @Jim: Can you clarify what you mean when you say Stembridge avoids type A? As far as I can see, Stembridge's notes solve a completely general problem. He is interested in the "unfolding" problem: given a root system Phi, find a simply-laced root system that folds onto Phi. Naturally, you wouldn't need this if Phi were simply-laced to begin with. | |
| Jul 31, 2013 at 0:14 | comment | added | Tom | Have you checked out two of Dyer's recent papers: 1) Embeddings of root systems. I. Root systems over commutative rings. J. Algebra 321 (2009), no. 11, 3226–3248. 2) Embeddings of root systems. II. Permutation root systems. J. Algebra 321 (2009), no. 3, 953–981. Not sure if you have seen those or if they are of interest, but thought I would mention them. | |
| Jul 30, 2013 at 21:38 | comment | added | Geordie Williamson | I found Lusztig's construction of embeddings using the $W$-graph of a left cell containing a simple reflection enlightening. See the end of "some examples of square integrable reps of p-adic groups". For example one gets natural embeddings H3 in D6 and H4 in E8 as well as more well-known examples. | |
| Jul 30, 2013 at 17:33 | answer | added | Jim Humphreys | timeline score: 4 | |
| Jul 30, 2013 at 8:18 | comment | added | Dima Pasechnik | Pasini's book is published in 1994 by Oxford U.P.: ukcatalogue.oup.com/product/9780198534976.do#.Ufd2n2TF0UI | |
| Jul 29, 2013 at 23:01 | history | edited | Ricardo Andrade |
added top level tag
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| Jul 29, 2013 at 20:16 | comment | added | Jim Humphreys | @userX: Thanks for the correction, which I've now made. Sometimes I type faster than I think. | |
| Jul 29, 2013 at 20:15 | history | edited | Jim Humphreys | CC BY-SA 3.0 |
deleted 1 characters in body
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| Jul 29, 2013 at 20:11 | comment | added | Jim Humphreys | @Dima: It would be helpful to add explicit references to material in Pasini's large book, which I haven't seen (it's not in any of our local college libraries). | |
| Jul 29, 2013 at 20:10 | comment | added | Jim Humphreys | @Johannes: Thanks for the reference to Stembridge's notes, which I had overlooked (though he avoids the case of type $A_n$ with $n$ even while making use of Lie theory). It's hard to define what's "in the literature", but the many lecture notes and such posted on home pages tend to be temporary and often unfindable in database searches. | |
| Jul 29, 2013 at 17:05 | comment | added | user36938 | A typo: the Weyl group of types B and C has $S_{\ell}$ acting on a normal subgroup $(\mathbf{Z}/(2))^{\ell}$ in the natural manner, not the other way around (as is written above, and doesn't make sense for $\ell > 2$). | |
| Jul 29, 2013 at 15:47 | comment | added | Dima Pasechnik | Foldings of geometries (including Coxeter geometries, i.e. these of finite reflection groups) are discussed in e.g. A.Pasini's book "Diagram geometries". | |
| Jul 29, 2013 at 15:03 | comment | added | Johannes Hahn | John Stembridge has written a few pages on such an elementary construction here: math.lsa.umich.edu/~jrs/papers/folding.ps.gz I don't know if that counts as "in the literature". | |
| Jul 29, 2013 at 13:51 | history | asked | Jim Humphreys | CC BY-SA 3.0 |