Timeline for When does Lusztig's canonical basis have non-positive structure coefficients?
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17 events
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| Jun 22, 2022 at 7:16 | history | edited | CommunityBot |
replaced http://front.math.ucdavis.edu/ with https://arxiv.org/abs/
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| Oct 16, 2010 at 0:06 | history | edited | Ben Webster♦ | CC BY-SA 2.5 |
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| Oct 3, 2010 at 22:54 | comment | added | David Hill | I thought you were suggesting that the $\{b_g\}$ basis might only coincide with the canonical basis up to sign, though as I understand you now, this is not the case. I do not know of any alternative to Saito's result. | |
| Oct 3, 2010 at 9:49 | comment | added | Hiraku Nakajima | There is NO issue. Lusztig proved that the base $\{ b_g \}$ coincides with the base given by perverse sheaves (and hence is the canonical base) for type $ADE$. And later Saito proved the same result in general. This result is true for any choice of a reduced expression of the longest element of the Weyl group. Since Saito's proof use the whole of Kashiwara's theory, I am looking for a simpler proof. | |
| Oct 3, 2010 at 1:08 | comment | added | David Hill | This is a very interesting point and maybe there is an issue? Leclerc constructs his basis $\{b_g\}$ using Lusztig's construction (so in some sense it is the canonical base by definition). First, he defines a PBW basis $\{E_g\}$ using an action of the braid group (there is more than one choice of action, but I don't know how much it matters). Then he obtains a unique bar invariant $b_g=E_g + \cdots$. The $E_g$ that Leclerc constructs are very natural in the sense that they have a well defined highest order term, and the coefficient of the leading terms are positive. | |
| Oct 1, 2010 at 21:56 | comment | added | Hiraku Nakajima | I do not know any results on this problem. By the way, could you explain how Leclerc (and you) concludes his base $\{ b_g\}$ coincides with the canonical base ? Easy to show that they coincide up to sign. But how does he remove the sign ambiguity ? The only way I know is to use Saito's result (Publ. RIMS, 1994). I would like to know if there are other methods. | |
| Oct 1, 2010 at 19:57 | comment | added | David Hill | No problem. I find this whole discussion very interesting. The Lyndon basis theory works really well for these algebras. It seems we have two different canonically defined bases coming from the same dual PBW basis. How are they related? Is there some way to twist the isomorphism in Kashiwara's definition of the global basis to make them coincide? Or is the relation more complicated? | |
| Sep 30, 2010 at 23:39 | comment | added | Hiraku Nakajima | In the last spring Kashiwara gave a talk on KLR algebras and asked us (in particular, to Tsuchioka) what is the base given by simple modules for non-symmetric cases. My comments above were vague, as it was not clear that I can make public Kashiwara's question. Sorry. | |
| Sep 30, 2010 at 16:18 | comment | added | David Hill | Okay, I changed it. | |
| Sep 30, 2010 at 16:17 | history | edited | David Hill | CC BY-SA 2.5 |
corrected a comment in light of a subsequent answer
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| Sep 30, 2010 at 11:43 | comment | added | Hiraku Nakajima |
Harry - Thank you. After Tsuchioka's answer, it is probably better to change everyone' to nobody'.
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| Sep 30, 2010 at 8:33 | comment | added | Ben Webster♦ | I always feel like one shouldn't need a result as strong as biadjointness, but this feeling keeps being wrong. | |
| Sep 30, 2010 at 8:32 | comment | added | Ben Webster♦ | David- You, of course, sussed out my reason for asking the question in the first place; I've seen Aaron Lauda assert in talks that outside symmetric type the basis coming from categorifications can't be Lusztig's because it has positive structure coefficients and Lusztig's doesn't. In symmetric type, it's known that you get the right thing for the enveloping algebra by Vasserot and Varagnolo's paper (arXiv:0901.3992). For reducing to cyclotomic quotients, you might have to use the stuff in my papers (which does include biadjointness of the E_i and F_i functors). | |
| Sep 30, 2010 at 7:54 | comment | added | Harry Gindi | I got it for you. | |
| Sep 30, 2010 at 7:53 | history | edited | Harry Gindi | CC BY-SA 2.5 |
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| Sep 30, 2010 at 3:35 | comment | added | Hiraku Nakajima | Could you exclude Nakajima from `everyone' in the 6th line, please ? | |
| Sep 29, 2010 at 23:44 | history | answered | David Hill | CC BY-SA 2.5 |