@MartinEscardo That would depend on how well the theorems, and more importantly, the definitions are written...

@julesh As a non-native English speaker who usually tries to approximate American English, I find this intriguing.

What do you mean by 'the "u" sound doesn't exist in American' - doesn't it occur in all of "dools, duals, jewels, duels"?

And which accent pronounces "duals" like "jewels"?

There is another way to view the sum of all irreps of G₂ as an algebra.

By Peter-Weyl, the space of smooth functions (complex-valued of finite type) on a compact Lie group G decomposes (under the left- and right-translation action) into a direct sum ⨁V*⊗V, where V runs through all complex […]

The space ⨁ₖV(k) has a geometric meaning. Think back to V(k) as a quotient of the symmetric power Symᵏ𝔰𝔬(7).

It turns out that the direct sum ⨁ₖV(k) can be realized as a quotient of the symmetric algebra over 𝔰𝔬(7), which (after dualizing with an invariant inner product) means a quotient of the […]

@andrejbauer @highergeometer I know U(1), but what's U₁(1)?

@highergeometer good suggestion; done!

@highergeometer Anytime I try to read copyright agreements, my mind just goes blank. Even when I genuinely want to understand them! It's like back in school when I tried (and failed) to study for history exams.

As a consequence, I must admit I usually don't really read nor understand the […]

@highergeometer what do the funny angular brackets mean?

@highergeometer @littmath @arXiv maybe it might be a good idea to allow users to flag articles in some way (perhaps with the option of adding a comment, only visible to moderators), so that the moderators can weed out nonsense much quicker and more reliably?

@highergeometer I think it's a standard fact that Λ²ℂⁿ is irreducible over SU(n), while Λ²ℂ²ⁿ=ℂω⊕Λ²₀ℂ²ⁿ over Sp(n).

If you need a reference for specific branchings anyway, try
"Tables of dimensions, indices, and branching rules for representations of simple Lie algebras" by McKay and Patera […]

@highergeometer Sorry for keeping you in suspense! My expertise on this stuff is rudimentary, so I don't think I can confidently write a quality MO answer here. In fact, I don't understand why π₃ should even be generated by a homomorphism from SU(2) - or whether every index 1 su(2)-subalgebra is […]

@highergeometer Right - if 𝔥⊂𝔤 with index 1 such that 𝔤/𝔥≅𝔥⊕…, then one can also take 𝔰𝔲(2)⊂𝔥 with index 1 and has 𝔤/𝔰𝔲(2)≅𝔰𝔲(2)⊕…, and the index multiplies to 1*1=1.

However I don't know enough Lie theory to ascertain that the index 1 𝔰𝔲(2)-subalgebras of a given 𝔤 are always generated by some […]

@highergeometer Maybe there really is a structural reason that this does not happen. On a related note, may I ask why π₃(𝐺/𝐻)=1 is important?