Timeline for Technical lemma on root systems, reduced to linear algebra
Current License: CC BY-SA 3.0
23 events
| when toggle format | what | by | license | comment | |
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| S Jan 27, 2017 at 16:32 | history | bounty ended | CommunityBot | ||
| S Jan 27, 2017 at 16:32 | history | notice removed | CommunityBot | ||
| Jan 26, 2017 at 0:59 | history | edited | Abhishek Parab | CC BY-SA 3.0 |
Added link to a newer question for the special case of G = SL(n).
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| Jan 20, 2017 at 15:11 | answer | added | Jim Humphreys | timeline score: 4 | |
| Jan 19, 2017 at 22:51 | comment | added | Abhishek Parab | Prof. Humphreys: I have made another edit justifying the case $\theta = 1$. I had the argument many days ago and in my mind but clumsily forgot to include it above, I apologize. | |
| Jan 19, 2017 at 22:48 | history | edited | Abhishek Parab | CC BY-SA 3.0 |
Added justification to the case $\theta = 1$ in the previous edit.
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| Jan 19, 2017 at 20:11 | comment | added | Jim Humphreys | Your edit is still out of focus, however, so please formulate the question more precisely. (Note that $\lambda - w \lambda$ is always in the positive root cone for dominant $\lambda$ but need not be in the dominant Weyl chamber.) | |
| Jan 19, 2017 at 19:53 | history | edited | Abhishek Parab | CC BY-SA 3.0 |
Made changes to improve clarity, cf. comments by Prof. Humphreys.
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| Jan 19, 2017 at 19:47 | comment | added | Abhishek Parab | Yes, $w = (1,2) = s_{\beta_1}$ is the 'first' simple reflection. I need the $\lambda$ to begin with, in the positive Weyl chamber (positive weight cone, if I may), i.e., having positive coefficients when expressed in a basis of weights. I'm not concerned with the positive root cone. The $\Omega$ should be subset of the positive Weyl chamber. Hope that clarifies. Apologies about the lemma numbering, edited. | |
| Jan 19, 2017 at 19:20 | comment | added | Jim Humphreys | Sorry to add another comment, but I'm still trying to understand your formulation-plus-edit. I guess the question only involves the root system (with Weyl group and weight lattice), along with an automorphism $\theta$ of the Dynkin diagram? (In your example, $W\cong S_3$, so is $w$ just the first simple reflection?) Usually the positive/dominant Weyl chamber is a proper subset of the positive root cone, so the proposition you quote from Bourbaki doesn't help if $\theta=1$ (it's actually in 1.6). Please clarify. | |
| S Jan 19, 2017 at 15:13 | history | bounty started | Abhishek Parab | ||
| S Jan 19, 2017 at 15:13 | history | notice added | Abhishek Parab | Draw attention | |
| Jan 18, 2017 at 3:08 | comment | added | Abhishek Parab | The main project is related to Arthur's twisted trace formula and the automorphism $\theta$ is the one which appears there. The expression (involving sums and integrals of truncated kernel) looks nasty but has a polynomial times $\exp \left( -\langle \lambda - \theta w \lambda, \varpi_\beta^\vee \rangle\right)$. The $\lambda$ is whose imaginary axis I would take Mellin transform and for correctly chosen $\lambda$, the negative exponential would dominate the polynomial, thus proving convergence. | |
| Jan 18, 2017 at 2:57 | comment | added | Abhishek Parab | $\Delta^B = \phi, \Delta = \Delta_B^G$ and whenever $P \subseteq Q, \Delta^P \subseteq \Delta^Q$. | |
| Jan 17, 2017 at 23:20 | comment | added | LSpice | I'm not being coy; I just don't know the answer (yet?). It's an interesting question; could you say anything about how you arrived at it? (Also, I always forget the convention; does $\Delta^B$ equal $\Delta$ or $\emptyset$?) | |
| Jan 17, 2017 at 17:07 | comment | added | Abhishek Parab | I would appreciate any comments / hints / ideas / thoughts from Prof. Humphreys or Loren. | |
| Jan 17, 2017 at 17:04 | comment | added | Abhishek Parab | That's right but I included it anyways. | |
| Jan 17, 2017 at 17:00 | comment | added | Jim Humphreys | The phrase "over a number field" isn't needed here, is it? The split assumption seems to be enough, since your question concerns just the Chevalley structure theory of such groups over arbitrary fields. | |
| Jan 17, 2017 at 15:46 | history | edited | Abhishek Parab | CC BY-SA 3.0 |
Added reference for the special case when $\theta=1$.
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| Jan 16, 2017 at 21:12 | comment | added | Abhishek Parab | Yes. (Otherwise it's not true.) | |
| Jan 16, 2017 at 20:22 | comment | added | LSpice | Is the cone allowed to depend on $w$? | |
| Jan 16, 2017 at 19:26 | history | asked | Abhishek Parab | CC BY-SA 3.0 |