- #1
Silviu
- 612
- 11
Hello! I am reading some things about representation theory for SU(n) and I want to make sure I understand it properly. I will put an example here and explain what I understand out of it and I would really appreciate if someone can tell me if it is right or not. So for SU(2) we have ##2 \otimes 2 = 1 \oplus 3##. Assume that we have a 2D object transforming under an irreducible representation (irrep) of SU(2) (I think this would be a spinor?) ##v= \begin{pmatrix}
a \\
b
\end{pmatrix}## and ##M \in SU(2)## such that ##v'=Mv##. Now ##2 \otimes 2## means we have 2 objects of this kind, so an element u here would be written as ##u = v \otimes v =
\begin{pmatrix}
a^2 \\
ab \\
ba \\
b^2
\end{pmatrix}##. The matrix N acting on this would be ##4 \times 4## and it can be reduced into something like this ## N =
= \begin{pmatrix}
x & 0 & 0 & 0 \\
0 & x_1 & x_2 & x_3 \\
0 & x_4 & x_5 & x_6 \\
0 & x_7 & x_8 & x_9
\end{pmatrix}##. Is this correct? So this means that ##a^2## transforms differently than the other 3 components of u? Thank you!
a \\
b
\end{pmatrix}## and ##M \in SU(2)## such that ##v'=Mv##. Now ##2 \otimes 2## means we have 2 objects of this kind, so an element u here would be written as ##u = v \otimes v =
\begin{pmatrix}
a^2 \\
ab \\
ba \\
b^2
\end{pmatrix}##. The matrix N acting on this would be ##4 \times 4## and it can be reduced into something like this ## N =
= \begin{pmatrix}
x & 0 & 0 & 0 \\
0 & x_1 & x_2 & x_3 \\
0 & x_4 & x_5 & x_6 \\
0 & x_7 & x_8 & x_9
\end{pmatrix}##. Is this correct? So this means that ##a^2## transforms differently than the other 3 components of u? Thank you!