Timeline for Quadratic Casimir of fundamental irreps of simply-laced Lie algebras
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Apr 6, 2016 at 0:47 | vote | accept | Peter Kravchuk | ||
| Sep 23, 2014 at 23:34 | answer | added | André Henriques | timeline score: 7 | |
| Jul 8, 2014 at 13:54 | comment | added | Jim Humphreys | @Peter: If you send me an email, I can explain in a little more detail the idea coming from Freudenthal's method which seems to be crucial in your observation (but only case-by-case so far): The underlying question is to relate the "support" of a simple root $\alpha_i$ in the set of all $N$ positive roots to the Coxeter number $h=2N/\ell$ when the corresponding fundamental weight $\varpi_i$ is minuscule. | |
| Jul 4, 2014 at 9:49 | vote | accept | Peter Kravchuk | ||
| Apr 6, 2016 at 0:47 | |||||
| Jul 2, 2014 at 13:55 | answer | added | Jim Humphreys | timeline score: 6 | |
| Jul 2, 2014 at 1:04 | history | edited | Peter Kravchuk | CC BY-SA 3.0 |
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| Jul 2, 2014 at 1:02 | comment | added | Peter Kravchuk | @JimHumphreys I think that the representations are indeed minuscule. I don't know a simple proof of this, but it seems to follow from what I know about the affine motivation of this question. From the motivation I also want to consider the trivial irrep, for which the equation is trivial. | |
| Jul 2, 2014 at 0:51 | comment | added | Peter Kravchuk | @JimHumphreys, I am looking at exactly those weights which are finite projections of affine dominant weights at level $k=1$. The normalization is that the long roots have square length 2. | |
| Jul 2, 2014 at 0:46 | comment | added | Peter Kravchuk | @JimHumphreys, yes, $g^\vee$ is the same as $h^\vee$. I am sorry, I mixed the notation from Di Francesco, he uses $g$ for dual Coxeter number, and ${}^\vee$ from Kac. Marks are defined as the coefficients of expansion of the highest root in the basis of simple roots, i.e. for $A_r$ all marks are $1$. I should probably say comarks (coming from the expansion in dual roots), but this is not important in simply-laced case. | |
| Jul 1, 2014 at 20:10 | comment | added | LSpice | @JimHumphreys, of course notation isn't the important thing, but maybe $g$ in place of $h$ comes from Suter's paper (the only other detailed discussion of dual Coxeter numbers that I've seen)? ams.org/mathscinet-getitem?mr=1600666 | |
| Jul 1, 2014 at 19:17 | comment | added | Jim Humphreys | P.S. To be more specific, are you looking at just the fundamental weights corresponding to minuscule representations? (These have just a single orbit of weights under the Weyl group, but don't exist for some types including $E_8$.) | |
| Jul 1, 2014 at 15:06 | comment | added | Jim Humphreys | @Peter: Can you clarify the formulation? For instance, your $g^\vee$ is called $h^\vee$ by Kac and coincides with the usual Coxeter number $h$ in simply-laced cases. Does your question have an analogue for other irreducible root systems? Also, what does "mark 1" mean? (And note that the old-fashioned symbol \varpi for pi, rather than \omega, was used for weights to suggest the French "poids".) Do you have a specific normalization of the inner product in mind (though it doesn't seem to matter here)? | |
| Jul 1, 2014 at 7:27 | history | edited | Peter Kravchuk | CC BY-SA 3.0 |
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| Jul 1, 2014 at 7:26 | history | migrated | from math.stackexchange.com (revisions) | ||
| Jul 1, 2014 at 7:25 | comment | added | Peter Kravchuk | @MarianoSuárez-Alvarez, I considered asking it there, but I though that it is likely that I am just not seeing something obvious here. If you think it's an Ok question for MathOverflow, I would be grateful if you do move. | |
| Jul 1, 2014 at 7:22 | comment | added | Mariano Suárez-Álvarez | I can move this question to MathOverflow, if you want. | |
| Jul 1, 2014 at 7:13 | history | asked | Peter Kravchuk | CC BY-SA 3.0 |