Timeline for highest weight representations inside tensor product
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Jan 12, 2015 at 20:17 | vote | accept | prochet | ||
| Jan 12, 2015 at 19:16 | history | edited | prochet |
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| Jan 12, 2015 at 18:23 | comment | added | Jim Humphreys | @prochet: By the way, the tag 'ag.algebraic-geometry' is also unneeded here and could be replaced by 'lie-algebras'. | |
| Jan 12, 2015 at 18:21 | answer | added | Jim Humphreys | timeline score: 5 | |
| Jan 12, 2015 at 14:19 | comment | added | Jim Humphreys | @prochet: I guess all you can say here is that $V(\lambda)$ is characterized as the unique direct summand of the tensor product involving the highest weight (in any direct sum decomposition into irreducibles). (Also, your tag 'lie-groups' should be replaced by 'algebraic-groups', given your assumptions.) | |
| Jan 12, 2015 at 13:36 | comment | added | Robert Bryant | @prochet: Identify $R$ with the sub-algebra of $U(\frak{g})$ that is generated by the so-called Casimir elements. Then the action of $U(\frak{g})$ on $V(\mu)\otimes V(\nu)$ is just the tensor product action; now restrict it to $R$. | |
| Jan 12, 2015 at 13:26 | comment | added | prochet | How $R$ acts on this tensor product? | |
| Jan 12, 2015 at 12:45 | comment | added | Robert Bryant | @prochet: I don't really understand what you mean by 'eliminate' in your comment. However, another possibility might be this: If $R\subset \mathsf{S}(\frak{g}^\ast)$ is the ring of $\mathrm{Ad}(G)$-invariant polynomials (for example, the Casimir would be an element of degree $2$), then $R$ acts as a commuting ring on $V(\mu)\otimes V(\nu)$, and you could, perhaps characterize $V(\mu{+}\nu)$ as the space of elements $s\in V(\mu)\otimes V(\nu)$ that satisfy $r(s) = h_{\mu{+}\nu}(r)s$ for the appropriate 'eigenvalue' homomorphism $h_{\mu{+}\nu}:R\to k$. | |
| Jan 12, 2015 at 10:27 | comment | added | prochet | for example there is a finite group acting on the tensor product wich eliminates everything but $V(\lambda)$. | |
| Jan 12, 2015 at 10:26 | history | edited | prochet | CC BY-SA 3.0 |
added 23 characters in body
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| Jan 12, 2015 at 9:10 | comment | added | Tobias Kildetoft | You seem to be assuming $k$ to have characteristic $0$, is this correct (in which case it would be a good idea to add it to the question). | |
| Jan 12, 2015 at 9:02 | comment | added | Vít Tuček | What kind of characterisation do you hope for? | |
| Jan 12, 2015 at 8:38 | history | asked | prochet | CC BY-SA 3.0 |