Timeline for Source of a formula for tensor product multiplicities?
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| May 20, 2013 at 12:20 | comment | added | Jim Humphreys |
P.S. The paper by King-Wybourne requires the added hypothesis that $V(\lambda)$ (in the above formula I was told about) is self-dual. This is true for the adjoint module in your special case, but I'm uncertain whether the added hypothesis is really needed or not. I wonder whether the newer methods make that clear?
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| May 19, 2013 at 16:42 | history | edited | Jim Humphreys | CC BY-SA 3.0 |
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| May 18, 2013 at 15:19 | history | edited | Jim Humphreys | CC BY-SA 3.0 |
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| May 18, 2013 at 14:42 | history | edited | Jim Humphreys | CC BY-SA 3.0 |
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| May 11, 2013 at 13:27 | comment | added | Jim Humphreys | @Allen: Yes, it seems reasonable to get this from the recent methods, but I'm curious about who first wrote this down (and where, why, how). There's a fairly extensive literature, hard to search: Dynkin, Kostant, Dixmier, Joseph, .... Kempf too had papers on tensor product decompositions, from the geometric viewpoint. | |
| May 10, 2013 at 12:43 | comment | added | Allen Knutson | Obviously I didn't know this more general result, but Evan Jenkins' modern proof generalizes easily enough. Restate as ${\rm Hom}(V(\lambda),{\mathfrak g}\otimes V(\lambda))$. Then we're looking at appending a $\Theta$-path of weight $0$ to a dominant $\lambda$-path, keeping the whole inside the dominant chamber (here $\Theta$) is the highest root. There are $rank(\lie{g})$ many $\Theta$-paths of weight $0$, of course, and the Lakshmibai-Seshadri ones go $0 \to -\alpha/2 \to 0$. | |
| May 9, 2013 at 15:58 | history | asked | Jim Humphreys | CC BY-SA 3.0 |