Gamma matrices out of pauli matrices - symmetry/group theory

[Otterhoofd]

I'm reading an article (http://prb.aps.org/abstract/PRB/v82/i4/e045122) but I have some problems understanding certain definitions. The authors have introduced a basis of certain bands (four) and then continue to give the transformation matrices of the symmetry operators. One (rotation) of them is given as:
[tex]R2=\sigma_1 \otimes\tau_3[\tex]
with "In the above, sigma acts in the spin basis and tau acts in the basis of P1+ and P2− subbands"

What does this product look like? Is it really a kronecker/direct product of the two matrices? I'm confused because they work in different bases. Or can I just do the kronecker product, resulting in i times:
0 0 1 0
0 0 0 -1
1 0 0 0
0 -1 0 0

In the appendix, gamma matrices are also defined in a similar way.

Can anyone point me in the right direction or give some insight on this? Thanks

 

[BigJDubs]

!Yes, the product you describe is indeed a Kronecker product of the two matrices. The Kronecker product is defined as the tensor product of two matrices. In this case, it simply means that you take the first matrix and multiply it by the second matrix element-wise. So, the resulting matrix would be:0 0 1 0 0 0 0 -1 1 0 0 0 0 -1 0 0 This means that each element from the first matrix is multiplied by each element from the second matrix.As for the gamma matrices, they are generally used to describe Dirac spinors, and the notation used in the article is very similar. The gamma matrices are related to the Pauli matrices (σ), and they act on a particular basis. In this case, the gamma matrices act on the spin basis, while the Pauli matrices act on the basis of P1+ and P2− subbands.