Timeline for What is the image of the half/full twist in the Hecke algebra, in the Kazhdan-Lusztig basis? What is the corresponding complex of Soergel bimodules?
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10 events
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| Oct 30, 2010 at 7:22 | comment | added | Geordie Williamson | I don't know of any software. The thing that makes n = 4 much harder than n = 3 is the fact that there are non-trivial (ie \ne 1) KL polynomials. I once worked out the complex for the half twist (but none of its powers) for n = 4 and remember not finding the answer particularly enlightening! If you do manage to get a conceptual picture of what is going on I'd be very interested to hear about it. | |
| Oct 30, 2010 at 2:44 | answer | added | Ben Webster♦ | timeline score: 4 | |
| Oct 30, 2010 at 1:00 | comment | added | Vivek Shende | Thanks Geordie. In fact I spent about the whole day, and not alone either, working out what $\Delta^k$ is for $n=3$, but I guess its description at the end is not all that bad. Do you know what it is for $n=4$? $n=5$? How far does it stay reasonable to write down? Is there any computer software to help me explore this question? | |
| Oct 29, 2010 at 22:31 | comment | added | Geordie Williamson | A quick answer is it's complicated ... expressing T_{w_0} in the KL basis involves inverse Kazhdan-Lusztig polynomials, which are again KL polynomials up to (-1)^{?}. Hence the complex corresponding to the half twist will be very complicated for n large enough (but for, eg n = 3 it is easily described). However, I seem to remember Khovanov mentioning that some nice stabilization happens in the Rouquier complex if one takes high powers of the full twist. I am not sure it has ever been written down. | |
| Oct 29, 2010 at 19:52 | history | edited | Vivek Shende | CC BY-SA 2.5 |
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| Oct 29, 2010 at 19:51 | comment | added | Vivek Shende | well, hopefully the link is improved. | |
| Oct 29, 2010 at 19:46 | history | edited | Vivek Shende | CC BY-SA 2.5 |
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| Oct 29, 2010 at 19:19 | comment | added | Jim Humphreys | I edited the names, but anyway the link to Wikipedia isn't right. | |
| Oct 29, 2010 at 19:18 | history | edited | Jim Humphreys | CC BY-SA 2.5 |
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| Oct 29, 2010 at 17:47 | history | asked | Vivek Shende | CC BY-SA 2.5 |