Timeline for Real and quaternionic representations according to weights
Current License: CC BY-SA 3.0
5 events
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Jun 8, 2015 at 11:05 | comment | added | Henrik Winther | @AllenKnutson I answered the question as written, maybe not as intended. I worry not only about the person that asked the question, who perhaps wants the compact case only, but also about others who might happen upon this because they want to know the answer to the question that was actually stated. (it's a very interesting question!) | |
Jun 8, 2015 at 10:27 | comment | added | Allen Knutson | All right, so your take on the question is about trying to extend the action of $G$ on $V$ to an action of $U(n,\mathbb R,J)$ vs. $U(n,\mathbb H,J)$ where the form $J$ need not be definite. Since the questioner made clear in comment that they're interested in the compact case, I believe this is an unnecessary wrinkle for them. | |
Jun 8, 2015 at 8:09 | comment | added | Henrik Winther | @AllenKnutson Consider the example I gave. Both have invariant skew symmetric forms, but one representation is real and the other quaternionic. So that cannot be the whole story. | |
Jun 8, 2015 at 2:25 | comment | added | Allen Knutson | I'm confused. I thought the issue was whether the invariant vector in $V\otimes V$ was in $Sym^2 V$ or $Alt^2 V$, and I don't see how this would be affected by choice of real form. | |
Jun 7, 2015 at 19:20 | history | answered | Henrik Winther | CC BY-SA 3.0 |