Bifundamental representations Can someone give me explicit examples (in matrix form) of bifundamental representations? Illustrative would be for instance:
a) SU(3) x SU(2)
b) SO(4) x U(1)
c) E6 x U(1)
but other you may have ready would also work. A reference would also be great.
 A: I will talk about $SU(3) \times SU(2)$.
First, a matrix $T_3 \in SU(3)$ acts in the fundamental representation on $\mathbb C^3$ in the following way: A vector $\vec v \in \mathbb C^3$ with components $v_i$ is mapped to $v'_i = (T_3)_{ij} v_j$. Similarly, a $T_2 \in SU(2)$ acts on $\vec w \in \mathbb C^2$ as $w'k = (T_2)_{kl} w_l$.
The bifundamental representation of $SU(3) \times SU(2)$ acts on $\mathbb C^6$, the only difficulty is the choice of basis. It is easiest to describe the action if we label the components of $\vec v \in \mathbb C^6$ as $v_{ik}$ for $i \in \{1,2,3\}$ and $k \in \{1,2\}$.
Then the element $(T_3, T_2) \in SU(3) \times SU(2)$ acts as
$$ v'_{ik} = (T_3)_{ij} (T_2)_{kl} v_{jl} \;. $$
If you really want to write this in matrix form, you first have to fix some order of the basis of $\mathbb C^6$. For example, we could write the components in as a vector
$$ (v_{11}, v_{12}, v_{21}, v_{22}, v_{31}, v_{32})^T \;. $$
After some thinking, it turns out that the group action can then be written as the following matrix:
$$ \begin{pmatrix}
 (T_3)_{11} (T_2)_{11} & (T_3)_{11} (T_2)_{12} & (T_3)_{12} (T_2)_{11} & \cdots & (T_3)_{13} (T_2)_{12} \\
 (T_3)_{11} (T_2)_{21} & (T_3)_{11} (T_2)_{22} & \ddots && \vdots \\
 \vdots &&&& \vdots \\
 (T_3)_{31} (T_2)_{21} & \cdots &&& (T_3)_{33} (T_2)_{22}
\end{pmatrix} \;. $$
