Timeline for What can be said about these tensor representations of $\mathrm{SL}(2)$?
Current License: CC BY-SA 4.0
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| Apr 5 at 11:36 | history | edited | YCor | CC BY-SA 4.0 |
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| Apr 5 at 11:31 | history | edited | Malkoun | CC BY-SA 4.0 |
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| Apr 4 at 1:18 | comment | added | Malkoun | I took an example to understand the Schur-Weyl duality and it finally made sense to me. Thank you! It is interesting that one can look at the decomposition you have mentioned as a representation of $SL(2, \mathbb{C})$ alone, or as a representation of $S_n$ alone. I know, this is why it is called "duality", but I had to work out an example to understand it. Nice. As far as this post goes, could you perhaps write your comment as an answer? I am interested in a specific example of $G$ and $H$, but I guess I can work it out now. Thank you! | |
| Apr 3 at 22:11 | comment | added | Malkoun | Yes, I don't understand the Schur-Weyl duality sufficiently well right now. I will spend some time on that then! Thank you. | |
| Apr 3 at 20:28 | comment | added | Will Sawin | Schur-Weyl duality lets us write $W$ as a sum of tensor products of irreducible representations of $SL_2$ and irreducible representations of $S_n$ in an explicit way. So your question about decomposition to irreducible representations of $SL_2$ reduces to calculating the rank of $\rho$ when restricted to these irreducible representation of $S_n$. This is clearly calculable explicitly in terms of $G$ and $H$ so the only thing to ask for is some kind of closed-form formula. I don't expect such a thing exists, although I can't disprove it. | |
| Apr 3 at 17:54 | history | edited | Malkoun | CC BY-SA 4.0 |
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| Apr 3 at 17:10 | history | asked | Malkoun | CC BY-SA 4.0 |