Timeline for Coxeter exchanges in non-reduced words
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 10, 2015 at 16:44 | comment | added | Christian Stump | (It might be better to move this into an email conversation.) | |
| Nov 10, 2015 at 16:43 | comment | added | Christian Stump | I also wouldn't know why the pieces should be balls or spheres, I'd rather expect not to be (I will try to construct a counterexample tonight). Finally, the "exists" means I wonder whether one can decompose an Artin subword complex $(Q,w)$ into subword complexes of the form $(Q,w1,S1) \sqcup ... \sqcup (Q,wm,Sm)$. Every element in $(Q,w)$ is in one of the pieces $(Q,wi,Si)$, but I would actually hope that the complete pieces are present. | |
| Nov 10, 2015 at 16:40 | comment | added | Christian Stump | There is no reduction for Artin words. I believe that the greedy product only depends on a word up to braid relations (as I think it only depends on the "colored inversion sequence" r1,s1(r2),s1s2(r3),... for a word s1,s2,s3,... with corresponding simple roots r1,r2,r3,...). | |
| Nov 10, 2015 at 15:53 | comment | added | Allen Knutson | So in this theory, you look at subwords that are Artin-reduced but not necessarily Coxeter-reduced? Is it clear that they should all have the same greedy product? I guess you can always decompose according to (greedy product, skip locations), but maybe the pieces won't be balls? I don't understand what "exists" means. | |
| Nov 10, 2015 at 12:19 | comment | added | Christian Stump | Do you happen to see whether the decomposition in #2 always exists? | |
| Nov 10, 2015 at 12:18 | comment | added | Christian Stump | You can also do that if you prefer. But since $(Q \setminus(S \setminus T),w,T)$ is "contained" in $(Q,w,T)$, you can as well send it to $(Q,w,T)$. I wanted to use the latter to emphasize that you can embedd everything into $(Q,w,\{\})$, and for an element in $(Q,w,\{\})$, you can recover all $S$ mapping onto this element. (Or am I overseeing some obvious problem?) | |
| Nov 10, 2015 at 11:04 | comment | added | Allen Knutson | Should that map in #3 be to $(Q\setminus (S\setminus T), w, T)$? | |
| Nov 10, 2015 at 10:10 | history | answered | Christian Stump | CC BY-SA 3.0 |