Timeline for Multiplication of extreme vector
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
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| Apr 20, 2010 at 2:51 | vote | accept | Shizhuo Zhang | ||
| Apr 20, 2010 at 2:51 | |||||
| Apr 20, 2010 at 2:51 | comment | added | Shizhuo Zhang | @Jim, thank you very much! Now I know,because the multiplicity is 1, and this needs Weyl character formula to compute! | |
| Apr 19, 2010 at 22:44 | comment | added | Shizhuo Zhang | @Jim, I have reformulated this problem above and point out the paper | |
| Apr 19, 2010 at 22:07 | comment | added | Jim Humphreys |
As Steven implies, the language is too fuzzy at times. Some features of the tensor product of modules over $\mathbb{C}$ are easy to describe in terms of weights, but detailed module structure gets very complicated. The solution by Shrawan Kumar of the old PRV Conjecture (Parthasarathy, Range Rao, Varadarjan) in Invent. Math. 93 (1988) is a sample of this. "Extremal" weights in the irreducible case are just the Weyl group conjugates of the highest weight (all have multiplicity 1), but in a tensor product what is extreme/extremal? (And which Joseph paper do you refer to?)
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| Apr 19, 2010 at 22:00 | comment | added | Shizhuo Zhang | @Sam, I will formulate the problem again. Sorry for mislead | |
| Apr 19, 2010 at 21:21 | comment | added | Steven Sam | What does $e_{w(\lambda + \nu)}$ mean? $e_{w\lambda} \otimes e_{w\nu}$ is a weight vector of $V_\lambda \otimes V_\nu$ of weight $w(\lambda + \nu)$... is that what you want? | |
| Apr 19, 2010 at 21:12 | comment | added | Shizhuo Zhang | Is there a equality(with coefficient)describing the relationship? | |
| Apr 19, 2010 at 21:11 | comment | added | Shizhuo Zhang | yes, this just follows from definition of tensor products of representations. But what is the relation of $e_{w\lambda}\otimes e_{w\nu}$ and $e_{w(\lambda+\nu)}$ | |
| Apr 19, 2010 at 20:53 | history | answered | Steven Sam | CC BY-SA 2.5 |