\ V_3 \longrightarrow V_1 \otimes
V_2                               \label{11xh3}$$

which in fact satisfies the intertwiner condition:

$$(\Pi _{V_1} \otimes \Pi_{V_2}) (g)   \ C^{12}_3 =
C^{12}_3 \Pi _{V_3} (g)          \label{11xh4}$$

This equation expresses the commutativity of the following diagram,

$$\begin{array}{llcll}
 & V_3    &  \stackrel{C^{12}_3}{\longrightarrow}  &  V_1\otimes V_2
&  \\ \Pi _3 & \downarrow  &   &   \downarrow  &  \Pi _1 \otimes \Pi
_2  \\      \label{11xh5}
 &  V_3  & \stackrel{C^{12}_3}{\longrightarrow}    &  V_1\otimes V_2
&   \end{array}$$

Thus from a theoretical point of view, [*the truncation process pertaining to the real-space RG is nothing but a tensor product decomposition of representations of the group*]{} ${\cal G}$. To make this point more explicit let us suppose that $\cal H$ and ${\cal
H'}$ are given as tensor products as:

$${\cal H}  = \otimes _1^N   V      \label{11xh6a}$$

$${\cal H'}  = \otimes _1^{N'}   V'     \label{11xh6b}$$

where $V$ and $V'$ are irreducible representation spaces of ${\cal G}$. Then the block method of the previous section applied to this case is equivalent to the tensor product decomposition:

$$V \otimes \stackrel{n_s}{\ldots } \otimes V
\longrightarrow V'      \label{11xh7}$$

In Eq. (\[11xh7\]) one is establishing that the irrep $\Pi _{V'}$ is contained in the tensor product of $N$ copies of the irrep $\Pi _V$.

Obviously, the 