Groups and representations

[Bobhawke]

I have a few questions:

1) The tensor product of two matrices is define by

$$A \otimes B =\left( {\begin{array}{cc} a_{11}B & a_{12}B \\ a_{21}B & a_{22}B \\ \end{array} } \right)$$

for the 2x2 case with obvious generalisation to higher dimensions.

The tensor product of two groups is defined by
$$G \otimes H = \ { (g,h) \vert g \epsilon G , h\epsilon H \ }$$

My question is this: Does the tensor product, defined in the first equation, of two matrix representations of groups A and B form a representation of the tensor product of the two groups as defined in the second equation? And if so, why?

A general element of the group AxB is (a,b), with the group operation defined in the obvious way ie
$$(a_1 , b_1) * (a_2 , b_2) = (a_1 a_2 , b_1 b_2)$$
My first instinct to form a matrix representation of the group AxB would be to take matrix reps of A and B and make a block diagonal matrix out of them. Then if I multiplied together two such matrices I would get exactly the multiplication law described above.

However, the tensor product of matrix reps of A and B gives something completely different. It is not clear to me that such a matrix should be a rep of the group AxB, or that it should obey the above multiplication law if we use matrix multiplication.

2) Assuming that the tensor product of matric reps of groups A and B gives a matrix rep of the group AxB, then:
If the matrix reps of A and B are in the the fundamental representation, does the matrix I get by taking their tensor product live in the fundamental representation of AxB?

Finally...
3) This all came from a statement in a book I am reading at the moment, namely:
"The electromagnetic current belongs to the singlet and adjoint representation of SU(N)"
where the eleectromagnetic current is
$$j_{\mu} = \psi_a \gamma_{\mu} \psi_b$$
where a and b are flavour indices transforming under the fundamental represenation of SU(N). It is not clear to me why this object should be in the "singlet and adjoint representation" of SU(N).



Thanks in advance.

 

[TFT]

1) Yes.
2) Not necessarily.
3) Follow the defination of adjoint representation. A quick way to see this: the current has no spinor indices like a and b, so it cannot belong to the fundamental represenation of SU(N).

 

[Bobhawke]

Do you know where I might find a proof of 1)?

Im having some trouble with 3). I am considering how something with two indices each transforming under the fundamental representation behaves. I get two exponentials, which I want to combine into one, so I am thinking I need to use the BCH formula. In the end I want this to look like

$$\exp{i \Lambda^a f^{a}_{bc}}$$

since the structure constants are the generators in the adjoints rep. But I can't see how to get this from the BCH formula.

 

[RedX]

I always thought 2 was true. For example, SO(4)=SU(2)xSU(2), and you characterize the irreducible representations of SO(4) by the fundamental representations of SU(2): (0,0), (1/2,0), (0,1/2), (1/2,1/2) ...

I think you're confusing some concepts of group theory. Take the direct product of groups G and H, GxH. If you stack G and H in block-diagonal form in a larger matrix, then this larger matrix does form a representation of the group GxH. However, in general, this representation will be reducible. An example is the fundamental representation of SU(5) under the subgroup SU(3)xSU(2)xU(1). This representation breaks up into (3,1,Y)+(1,2,Y'), where Y and Y' is the U(1) hypercharge which I can't remember.

edit: o wait, for 2) you said fundamental representation, not irreducible representation. oops. actually come to think of it I don't know what fundamental representation means when it comes to direct products of groups.

 

[Bobhawke]

Ok I can show that $$F_{\mu \nu}= F_{\mu \nu}^a t^a$$ transforms under the adjoint representation, in the sense that one of the contracted indices a gets acted on by a matrix in the adjoint rep (which is kinda weird I think), but the proof relies on the fact that it is contracted with the generators. I can't figure out why $$\psi_a \psi_b$$ would transform under the adjoint because there are no generators kicking about, so the proof I did seems not to work.

RedX, when I said fundamental rep of AxB I guess I assumed that this would be isomorphic to a group whose fundamental rep would be defined.