Difference between revisions of "Talk:Aufgaben:Problem 13"
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| − | Maybe a little shortening. Instead taking \(S^{(k,l)}\) and \(T^{(k,l)}\), take a more generall \(U^{(k,l),[\rho,\sigma]}_{ij}=(\rho_{ki},\sigma_{lj})_G\) for any | + | Maybe a little shortening. Instead taking \(S^{(k,l)}\) and \(T^{(k,l)}\), take a more generall \(U^{(k,l),[\rho,\sigma]}_{ij}=(\rho_{ki},\sigma_{lj})_G\) for any two unitary representations \(\rho, \sigma \) and with \(U^{(k,l),[\rho,\sigma]}\sigma(g)=\rho(g) U^{(k,l),[\rho,\sigma]}\) also \([T^{(k,l)},\rho(g)]=0\) is showed. |
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| + | \(\sum\limits_{i,j}{U^{(i,j),[\rho,\sigma]}_{i,j}}=(ch(\rho),ch(\sigma))_G\) may also save some time | ||
--[[User:Brynerm|Brynerm]] ([[User talk:Brynerm|talk]]) 17:01, 4 August 2015 (CEST) | --[[User:Brynerm|Brynerm]] ([[User talk:Brynerm|talk]]) 17:01, 4 August 2015 (CEST) | ||
Revision as of 17:42, 4 August 2015
There is an alternative way of proving (a) in the Felder Script:
https://people.math.ethz.ch/~felder/mmp/mmp2/
see chapter 3 -> Satz 3.1. and chapter 2 -> Satz 2.6 for a prove of Schur's lemma.
It seems to be shorter, and thus is probably better for the exam... Carl (talk) 23:16, 13 June 2015 (CEST)
Did anyone succeed in getting a shorter solution for a) by using Schur's lemma (as suggested by Carl)?
- Snick (talk) 13:06, 30 July 2015 (CEST)
I don't think you can make it any shorter. Shur's Lemma is used implicitly in the proof (claim 2), and to it you must first show that T is a homomorphism of representations (i.e. claim 1). The Felder Skript provesClaim 2 and Claim 5, but then you must still show the rest.
- Rayan (talk) 11:36, 31 July 2015 (CEST)
Maybe a little shortening. Instead taking \(S^{(k,l)}\) and \(T^{(k,l)}\), take a more generall \(U^{(k,l),[\rho,\sigma]}_{ij}=(\rho_{ki},\sigma_{lj})_G\) for any two unitary representations \(\rho, \sigma \) and with \(U^{(k,l),[\rho,\sigma]}\sigma(g)=\rho(g) U^{(k,l),[\rho,\sigma]}\) also \([T^{(k,l)},\rho(g)]=0\) is showed.
\(\sum\limits_{i,j}{U^{(i,j),[\rho,\sigma]}_{i,j}}=(ch(\rho),ch(\sigma))_G\) may also save some time
--Brynerm (talk) 17:01, 4 August 2015 (CEST)
Very nice, that does same some time.
