Lie groups and Lie algebras Optional Extra: Proof of classification of SU(3) irreps
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- čas přidán 9. 07. 2024
- We start the proof of the classification theorem of irreducible SU(3) representations, by showing that if V is an SU(3) representation and v is a highest weight vector then v generates a subrepresentation. This video is optional.
amazing video series, you've made it so easy!
I have learned from other sources that a decomposition of your representation space into your invariant subspaces U can be given, and in quite relevant terms. Namely, for any induced representation
R_* : *sl*(3, ℂ) → *gl*(Sym^a(V) ⊗ Sym^b(V*)),
the representation space decomposes as ⊕_{i=0}^b Γ_{a-i, b-i}.
Could you make a connection between these two approaches? Is your U one of these Γ_{a-i, b-i}?
I'm not sure I understand your claim (are you saying this is true for any V? Or just the standard rep?). But if you do have a representation with decomposition \bigoplus_{i=0}^b \Gamma_{a-i,b-i} then the argument I gave will pick out the copy of \Gamma with the highest weight vector (so \Gamma_{a,b}). Then you take the orthogonal complement of that subrep and continue, which will give the next \Gamma_{a-1,b-1}, etc etc etc
@@jonathanevans27 Ah yes, sorry, this was all for the representation space V = ℂ³. Then yes, I agree (this is basically Maschke's Theorem, if I'm correct). I am just getting a little confused over what the significance of Sym^a(V) ⊗ Sym^b(V*) is, especially pertaining to the quark model? Since V ⊗ V ⊗ V (for baryons) is not of this form..? Yet I've seen it claimed that Sym^a(V) ⊗ Sym^b(V*) is, in some sense, the most general rep. space we need to consider? (You _do_ note, incidentally, that Γ₃₀ = Sym³(ℂ³) = Sym³(ℂ³) ⊗ Sym⁰(ℂ³)...?)
For instance: _"From these two results we can deduce that the symmetric powers of the two fundamental representations must be irreducible and that all the representations which possess a triangular weight diagram must be some symmetric power of either the normal or the dual fundamental representation."_
@@jonathanevans27 Yes I do now realise how ill-posed the question was. This tensor product of symmetric powers of ℂ³ and ℂ³* are used specifically to accomodate quantum mechanical symmetries (including antiquarks), I think.
By the way, I handed in my thesis yesterday, and wanted to thank you personally for all your help. I hope we’ll meet some day, Lancaster is not that far, and I’m certainly prepared to leave my country (the NL) for a PhD position ;)
No but really, thanks a lot. Due to the corona pandemic, I was mainly confined to… myself. And the further you get in mathematics, the fewer people can help you 😅
Great content!
Bye!
Jos van der Spek
Why does the proof show it is irreducible, not just a subrepresentation?
@Jaco Van Tonder - You're right, it doesn't. That was left as a guided exercise (question 8 on sheet 4: jde27.uk/lgla/questions4.html)
Is this because no actual decomposition of U is given?