Timeline for Are the Weyl modules projectives?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Jun 20, 2012 at 12:15 | answer | added | Wilberd van der Kallen | timeline score: 4 | |
| Jun 20, 2012 at 0:39 | answer | added | George McNinch | timeline score: 6 | |
| Jun 19, 2012 at 21:16 | comment | added | Jim Humphreys |
@Chuck: I agree that the question is difficult to parse, but I don't agree that the terminology "Weyl module for the first Frobenius kernel" is used. Those highest weight modules (of fixed $p$-power dimension) usually get called something like "baby Verma modules" and are compared in subtle ways to the actual Weyl modules for the algebraic group. Their dimensions are usually quite different and they are constructed in different settings.
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| Jun 19, 2012 at 17:32 | vote | accept | Binai | ||
| Jun 19, 2012 at 15:59 | comment | added | Chuck Hague | As Jim notes below, this question is, unfortunately, difficult to parse because you're using nonstandard terminology. For example, "Weyl module for $\mathfrak g$" does not mean the same thing as "Weyl module for the hyperalgebra in positive characteristic." I would interpret "Weyl module for $\mathfrak g$" as "Weyl module for the first Frobenius kernel," in which case there are things you can say about projectivity; I would direct you to Jantzen's book "Representations of Algebraic Groups" for more on projective modules for Frobenius kernels. | |
| Jun 19, 2012 at 15:41 | answer | added | Jim Humphreys | timeline score: 7 | |
| Jun 18, 2012 at 17:12 | comment | added | Binai | Exactly, I am talking about this highest weight modules. In characteristic zero they are identical with the finite dimensional simple modules. In positive characteristic I refer to the Weyl module for the hyperalgebra (the tensor product over $\mathbb Z$ of an integral form of $\frak g$ with an algebraically closed field of characteristic $p$). However, I am also interested on the analogues of Chari for loop algebras! | |
| Jun 18, 2012 at 13:36 | comment | added | Jim Humphreys |
There is some confusion about exactly what you mean here by "Weyl module for $\mathfrak{g}$". The term "Weyl module" was invented by Carter-Lusztig to deal with situations in prime characteristic where you have a finite dimensional highest weight module (for an algebraic group) which fails to be completely reducible. There are also analogues in the work of Chari and others in different settings. In characteristic 0, are your "Weyl modules" identical with the finite dimensional simple modules? And what do you refer to in positive characteristic?
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| Jun 15, 2012 at 18:20 | history | edited | Binai | CC BY-SA 3.0 |
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| Jun 15, 2012 at 18:05 | answer | added | Bruce Westbury | timeline score: 1 | |
| Jun 15, 2012 at 17:59 | history | asked | Binai | CC BY-SA 3.0 |