Reladenine Vakalwe
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Registered User
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An evident pseudonym
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Jan 7 |
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Around the socle filtration of a Verma module Also, is Jantzen's Habilitationsschrift a reference to "Moduln mit einem hochsten Gewicht"? That text strikes fear in my heart! |
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Jan 7 |
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Around the socle filtration of a Verma module To be honest, apart from being slightly open to "miracles sometimes occur", I am pretty skeptical about a). I do believe in b) though because it matches with some heuristics I have regarding higher extensions between Vermas (heuristics coming from some brute force computations, but I haven't managed to work out all the extensions in Boe's counterexample, so these may still be "too low rank"). I think I understand Mazorchuk's proof (regarding b)) enough to see that he defers the burden to a result of Backelin. But haven't got my hands on the Backelin paper yet. |
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Jan 6 |
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Around the socle filtration of a Verma module I agree with the type $A_2$ check (although I don't think I could typeset the lattice/diagram). Stroppel's diagram is pretty impressive! I am reasonably sure that b) is true in general (since I believe Mazorchuk's result), but of course I don't understand why. |
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Jan 6 |
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Around the socle filtration of a Verma module Inspired by your example, as long as I did it correctly, c) is false in type $A_2$ also, and I am reasonably sure is essentially always going to fail (with the exception of $\mathfrak{sl}_2$). |
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Jan 6 |
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Around the socle filtration of a Verma module added 56 characters in body |
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Jan 6 |
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Around the socle filtration of a Verma module added 116 characters in body |
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Jan 6 |
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Around the socle filtration of a Verma module Yes! This is a nice counterexample. Not simple, but that's a minor quibble. Thank you! Any thoughts on b)? |
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Jan 6 |
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Around the socle filtration of a Verma module Dag: Could you elaborate on your second comment regarding the counterexample to c) (possibly as an answer)? |
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Jan 6 |
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Around the socle filtration of a Verma module 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? |
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Jan 6 |
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Around the socle filtration of a Verma module added 7 characters in body; deleted 93 characters in body |
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Jan 6 |
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Around the socle filtration of a Verma module 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! |
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Jan 6 |
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Around the socle filtration of a Verma module edited body |
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Jan 6 |
asked | Around the socle filtration of a Verma module |
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Dec 31 |
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(geometric/intuitive) interpretation of ext Not quite in line with your question. But if you were dealing with a reasonable topological space $X$, then $Ext$ groups of the constant sheaf with itself (in the category of constructible sheaves) are the cohomology groups of that space. More generally, extensions from the constant sheaf to any complex of sheaves is hypercohomology with coefficients in the complex. |
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Dec 31 |
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extensions of IC sheaves added 15 characters in body |
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Dec 31 |
answered | extensions of IC sheaves |
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Dec 31 |
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Morphisms between Verma modules As the Mathoverflow bot has so graciously pinged this question, I may as well add the following. The "curious Poincare duality" and "palindromic phenomenon" mentioned above has a high powered explanation. Namely: Koszul duality. This can in turn be used to show the dimension bound asked for. But this is using a blowtorch to light a candle. |
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Dec 29 |
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Splitting of the weight filtration Dan: Thanks for the examples! They are helpful. I wasn't aware of Minhyong Kim's paper. It's really nice. |
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Dec 29 |
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Splitting of the weight filtration Donu: Thanks! Your answer is helpful. At least how I am reading it is that the weight filtration has geometric as well as linear algebraic content to it (split over $\mathbb{R}$ vs. $\mathbb{Q}$). As an aside, related to your comment about varieties coming from linear algebra, most of my examples come from flag varieties and "miracles often happen in flag varieties"! |
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Dec 28 |
asked | Splitting of the weight filtration |
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Dec 21 |
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Geometric interpretation of translation through the wall Jim: Perhaps I should make this a separate question. But don't Soergel's arguments (which I am implicitly using to justify my answer below) show this? Under Soergel's functor to combinatorics $\mathbb{V}$, translation across the wall corresponds to (roughly) restriction/induction for the coinvariant algebra. The latter depending only on the stabilizer (namely $s$) of $\lambda$. Or am I confused? Of course, $\mathbb{V}$ is not an equivalence but it is full and faithful on maps between projectives/tiltings, but this should be enough to show the desired independence? No? |
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Dec 20 |
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Geometric interpretation of translation through the wall deleted 118 characters in body; deleted 1 characters in body |
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Dec 20 |
answered | Geometric interpretation of translation through the wall |
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Dec 20 |
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Non-characteristic is to pullback as (blank) is to pushforward. Slightly confused about a certain point, why is pushforward along a proper map preserving t-structure? Regarding the question, there is a good estimate on singular support of $f_*M$ if $f_{\pi}\colon f_d^{-1} Ch(M)\to T^*Y$ is finite (I hope the notation is self-explanatory. See Kashiwara's D-Modules and microlocal calculus section 4.7. This condition is analogous to the non-characteristic condition. |
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Dec 17 |
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Morphisms between Verma modules Deleted a previous comment of mine where I thought I had an argument, since it was a pipe dream. |
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Dec 16 |
awarded | ● Nice Question |
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Dec 14 |
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Morphisms between Verma modules edited body |
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Dec 14 |
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Morphisms between Verma modules added 2673 characters in body |
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Dec 14 |
asked | Morphisms between Verma modules |
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Dec 4 |
asked | Computing rational cohomology of smooth (not necessarily compact) toric varieties |
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Dec 3 |
awarded | ● Nice Answer |
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Dec 2 |
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A cohomology computation request. Ah, perfect! Many thanks! This is exactly the sort of simple answer I was looking for. |
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Dec 2 |
asked | A cohomology computation request. |
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Dec 1 |
answered | Non-rigorous reasoning in rigorous mathematics |
