Timeline for Kazhdan-Luzstig Polynomials and Lower Intervals in the Bruhat Order
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Feb 10, 2013 at 23:40 | comment | added | Jim Humphreys | @Vlad: My answer (and cross-reference to my later question with a citation of work by Carrell and Peterson, is probably optimal for your question. I agree with Alexander Woo's second sentence, though it's always possible that a shortcut might be found. | |
| Dec 8, 2012 at 12:35 | answer | added | Jim Humphreys | timeline score: 2 | |
| Jul 20, 2012 at 17:57 | answer | added | xpilot | timeline score: 0 | |
| Jul 14, 2012 at 17:37 | comment | added | xpilot | @Christian: I'm using a C++ program called "coxeter", written a while ago by Fokko du Cloux (math.univ-lyon1.fr/~ducloux/coxeter/coxeter3/english/…). The code is a bit outdated (doesn't use standard C++ data structures) but otherwise pretty clean and fast. | |
| Jul 14, 2012 at 7:56 | comment | added | Christian Stump | @Vlad: I wonder how you did these computations that fast - it would be nice if you could say some words about that. | |
| Jul 13, 2012 at 23:05 | comment | added | Alexander Woo | @Vlad: To your question about Jim, yes they are the same person. To speculate, I doubt this is any easier than nonnegativity. Perhaps it might be possible to prove this assuming nonnegativity, but of course that is cheating since nonnegativity also requires geometry to prove. And, to give advice, I think you get further in this subject if you at least know enough of the algebraic geometry to understand the basic ideas coming from that side of the subject, even if you can't master the machinery well enough to use it to prove anything. The same can probably be said of the rep theory. | |
| Jul 13, 2012 at 21:29 | comment | added | xpilot | Ah, so Jim Humphreys (mathoverflow user) = James Humphreys (author)? In that case, there probably isn't a known Coxeter-theoretic proof. I was hoping that such a proof would help with some related research I'm doing; maybe I'll switch to trying to find a proof myself. | |
| Jul 13, 2012 at 17:26 | comment | added | Christian Stump | @Vlad: as you mention yourself above, it is known for crystallographic types anyway (these are An,Bn,Dn,E6,E7,E8,F4,G2). So the only remaining (finite) non-crystallographic types are H3, H4, and the dihedral groups (see e.g. Jim Humphreys' book). So knowing now that the statement holds in all finite types gives hope that there is a Coxeter-theoretic proof. | |
| Jul 13, 2012 at 17:13 | comment | added | xpilot | @Qiaochu: I've checked it by computer for F4, H3, H4, A1-7, B1-6, D1-6, E6, E7. Not sure what else I'd check, aside from larger groups (which tend to take forever and cause memory overflows). @Jim: Most of what I know about Coxeter groups and Bruhat order is from Bjorner and Brenti's "Combinatorics of Coxeter Groups", and my main source for Hecke Algebra and KL-polynomials is James Humphreys' "Reflection Groups and Coxeter Groups". | |
| Jul 13, 2012 at 16:02 | comment | added | Jim Humphreys |
@Vlad: It would help to add a reference or two (maybe the 2009 Annals paper by Bjorner-Ekedahl?) and/or a bit more background. There are other open problems about non-crystallographic Coxeter groups and KL polynomials, e.g., the KL conjecture that coefficients of the polynomials should be non-negative. In the crystallographic case they can be interpreted as dimensions of cohomology groups. Alvis checked by computer the $H_3, H_4$ groups, while Soergel has proposed an ambitious program to show non-negativity without algebraic geometry. Your problem also looks very hard.
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| Jul 13, 2012 at 9:47 | answer | added | Christian Stump | timeline score: 3 | |
| Jul 13, 2012 at 1:55 | comment | added | Qiaochu Yuan | Have you tried looking for a counterexample in a non-Weyl group? | |
| Jul 13, 2012 at 1:06 | history | asked | xpilot | CC BY-SA 3.0 |