Timeline for How to compute the index of a given weight?
Current License: CC BY-SA 3.0
11 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
Commonmark migration
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| Jan 5, 2018 at 18:02 | comment | added | Vít Tuček | I've edited my answer to address your concerns. Sorry for negligence. | |
| Jan 5, 2018 at 18:01 | history | edited | Vít Tuček | CC BY-SA 3.0 |
fixes & expansion
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| Jan 5, 2018 at 1:09 | comment | added | Jim Humphreys | @LSpice: Sorry, I misread the proposed criterion by Vit. In any case, I can't sort things out further without knowing at least the numbering convention for simple roots. | |
| Jan 4, 2018 at 19:05 | comment | added | LSpice | (Excuse me; I meant to say "maybe I have not understood the condition".) | |
| Jan 4, 2018 at 17:02 | comment | added | LSpice | @JimHumphreys, indeed, my example referred to writing $\lambda = 1\epsilon_1 + 0\epsilon_2$ in $\mathsf B_2$, which (it seems to me) is singular even though, as required, the coefficients are pairwise distinct. (As you say, maybe I have understood the condition imposed.) I just added the expression in terms of the $\alpha$'s to check whether I understood correctly what the "Bourbaki $\epsilon$-basis" was. | |
| Jan 4, 2018 at 14:42 | comment | added | Jim Humphreys | @Vit: In the second question, note that the weight is assumed to be nonsingular, though "index" makes sense in all cases. | |
| Jan 4, 2018 at 14:40 | comment | added | Jim Humphreys | @LSpice: I agree with you that the wording of Vit's answer is sometimes imprecise; but in the first question his notion of "coefficient" refers to writing $\lambda$ as a linear combination of the $\epsilon_i$. In either question, I still wonder if there is a more efficient way to do such computations in general. | |
| Jan 4, 2018 at 11:46 | comment | added | LSpice | Also, you say that the index (of a weight) equals the length (I guess also of a weight), but what is the definition of the length of a weight? The post defines the index of a weight to be the length of a Weyl-group element bringing it into the fundamental chamber, but it seems that the poster doesn't want to compute it that way. (Are you saying that simply checking the definition, perhaps by counting walls, is the best way to compute it?) | |
| Jan 4, 2018 at 11:43 | comment | added | LSpice | What is "Bourbaki's $\epsilon$-basis"? If the one in the Lie-groups book, then I think your answer to (1) is not true. For example, it seems to me that, in $\mathsf B_2$, in Bourbaki's numbering, we have that $\lambda = \epsilon_1 = \alpha_1 + \alpha_2$ satisfies your condition, but $\langle\lambda, \alpha_2\rangle = 0$. | |
| Jan 4, 2018 at 10:11 | history | answered | Vít Tuček | CC BY-SA 3.0 |