Timeline for Around the socle filtration of a Verma module
Current License: CC BY-SA 3.0
15 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Jan 6, 2013 at 23:39 | answer | added | Jim Humphreys | timeline score: 3 | |
| Jan 6, 2013 at 19:19 | history | edited | Reladenine Vakalwe | CC BY-SA 3.0 |
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| Jan 6, 2013 at 18:35 | history | edited | Reladenine Vakalwe | CC BY-SA 3.0 |
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| Jan 6, 2013 at 17:15 | answer | added | Dag Oskar Madsen | timeline score: 2 | |
| Jan 6, 2013 at 16:30 | comment | added | Dag Oskar Madsen | Yes, I think it is clear now, it was just the "Note" that confused me. | |
| Jan 6, 2013 at 16:11 | comment | added | Reladenine Vakalwe | Dag: Could you elaborate on your second comment regarding the counterexample to c) (possibly as an answer)? | |
| Jan 6, 2013 at 16:10 | comment | added | Reladenine Vakalwe | Dag: Your first comment about the statement after "Note:" in a) being wrong is absolutely correct. I have removed it (sorry strikethrough wasn't showing up correctly). As to how I am choosing $k$. Basically I just want the first layer that isn't completely contained in $\Delta_w$. Does that clarify or am I being screwy? | |
| Jan 6, 2013 at 16:04 | history | edited | Reladenine Vakalwe | CC BY-SA 3.0 |
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| Jan 6, 2013 at 16:02 | comment | added | Reladenine Vakalwe | Jim: Thanks for pointing out the misprint. I have fixed it now. I agree about the risk of relying on rank 2 examples. But well, it's a start. Figuring out socle filtrations for higher rank is quickly going to start being a pain! Apologies for the length. I had hoped that getting a) out there quickly would alleviate some of the pain. My next step is to ask Mazorchuk, but am still holding out hope that I am missing something silly and MO will offer some instant gratification! | |
| Jan 6, 2013 at 15:55 | history | edited | Reladenine Vakalwe | CC BY-SA 3.0 |
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| Jan 6, 2013 at 15:11 | comment | added | Dag Oskar Madsen |
Statement (c) seems to be wrong when the algebra is $\mathfrak{so}_4$ and $\Delta_w$ is one of the length $2$ Vermas. On the other hand $\mathfrak{so}_4$ isn't simple. Maybe this could be used to get a counterexample for $\mathfrak{so}_6$?
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| Jan 6, 2013 at 14:25 | comment | added | Jim Humphreys |
@RV: It might take a while to read all of this (which makes my own questions look really short). But note the small misprint at the end of the first paragraph: "for all $w$" should be "for all $v$". More important, it's risky to rely on examples of rank 2, where K-L polynomials for the (dihedral) Weyl group are trivial. Early work of people like Jantzen, Gabber-Joseph, Irving shows how delicate the socle and Ext questions can be; so it's important to focus your own questions as tightly as possible. It might also be ueful to consult Mazorchuk directly.
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| Jan 6, 2013 at 13:35 | comment | added | Dag Oskar Madsen | I am a little confused about how you choose $k$, but I think the statment in the \bf{Note:} must be wrong since the socle layer might contain composition factors from $\Delta_w$. | |
| Jan 6, 2013 at 3:13 | history | asked | Reladenine Vakalwe | CC BY-SA 3.0 |