# Recent content by nigelscott

1. ### A Pullback of the metric from R3 to S2

I am looking at this document I do not understand how the author gets 5.12 and 5.13 on page 133. I think the matrix of partials should be the transpose of the one shown. Even so I still can't figure out how you get 5.13. Any help would be appreciated.
2. ### I 2 and 3 dimensional invariant subspaces of R4

I am looking at the representation of D4 in ℝ4 consisting of the eight 4 x 4 matrices acting on the 4 vertices of the square a ≡ 1, b ≡ 2, c ≡ 3 and d ≡ 4. I have proven that the 1-dimensional subspace of D4 in ℝ2 has no proper invariant subspaces and therefore is reducible. I did this in 2...
3. ### Lie Bracket and Cross-Product

OK. Thanks to you both. I think I understand it now.
4. ### Lie Bracket and Cross-Product

Ok. Thanks for your response. The example I am using is from this video here starting at 12 mins and continuing here. Here he talks about tangents to the sphere with the Lie bracket being another tangent to the sphere which is at odds with the cross product which would produce a vector normal...
5. ### Lie Bracket and Cross-Product

Prove that for a 2 sphere in R3 the Lie bracket is the same as the cross product using the vector: X = (y,-x,0); Y = (0,z-y) [X,Y] = JYX - JXY where the J's are the Jacobean matrices. I computed JYX - JXY to get (-z,0,x). I computed (y,-x,0) ^ (0,z,-y) and obtained (xy,y2,yz) = (z,0,x)...
6. ### A Block Diagonalization - Representation Theory

This will take me a little time to digest, but in the meantime I wanted to thank everyone for your responses.
7. ### A Block Diagonalization - Representation Theory

OK. Thanks. It seems that this involves numerical analysis and is best solved using matlab etc. Is that a fair assessment?
8. ### A Block Diagonalization - Representation Theory

How does one go about finding a matrix, U, such that U-1D(g)U produces a block diagonal matrix for all g in G? For example, I am trying to figure out how the matrix (7) on page 4 of this document is obtained.
9. ### I Block Diagonal Matrix and Similarity Transformation

I am looking at page 2 of this document.https://ocw.mit.edu/courses/chemistry/5-04-principles-of-inorganic-chemistry-ii-fall-2008/lecture-notes/Lecture_3.pdf How is the transformation matrix, ν, obtained? I am familiar with diagonalization of a matrix, M, where D = S-1MS and the columns of S...
10. ### A Decomposition of tensors into irreps (Georgi's book)

Perfect. Thank you!
11. ### A Decomposition of tensors into irreps (Georgi's book)

Hello. I just came across this post. I understand how the 2nd term in (4) is to equivalent to the 1st term in the 1st equation i.e. uivjk - ujvik = εijlεlmnunvmk = (δimδjn - δjnδim)umvnk but I am having a mental block regarding how the first term in (4) is equivalent to the 2nd term in the...
12. ### Clebsch-Gordan Decomposition for 6 x 3

Yes, I am familiar and recognize that Young tableaux is easier for practical purposes. However, this approach should give the same result, yes?

SU(3).
14. ### Clebsch-Gordan Decomposition for 6 x 3

Thanks. Yes, I am familiar with using the ladder operators. I was more focused on the procedure outlined using the 3 ⊗ 8 by Georgi (LIe Algebras in Particle Physics page 143) and also here https://physics.stackexchange.com/questions/102554/tensor-decomposition-under-mathrmsu3 . I was trying...
15. ### Clebsch-Gordan Decomposition for 6 x 3

Homework Statement [/B] I am trying to get the C-G Decomposition for 6 ⊗ 3. 2. Homework Equations Neglecting coefficients a tensor can be decomposed into a symmetric part and an antisymmetric part. For the 6 ⊗ 3 = (2,0) ⊗ (1,0) this is: Tij ⊗ Tk = Qijk = (Q{ij}k + Q{ji}k) + (Q[ij]k +...