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I am struggling to find the correct concept for concatenating representations of different groups, in a block matrix format. Let us consider two groups, $G_1$ and $G_2$, as well as two finite-dimensional representations $\rho_1$ and $\rho_2$ respectively of $G_1$ and $G_2$ (of dimensionalities resp. $n_1$ and $n_2$). Let us consider the mapping $\rho: G_1 \times G_2 \to GL(n_1 + n_2)$ defined by:

$$\rho((g_1, g_2)) = \begin{pmatrix} \rho_1(g_1) & 0 \\\\ 0 & \rho_2(g_2)\end{pmatrix}.$$

If my understanding is correct, it is easy to show that $\rho$ is in this case a representation of the direct product $G_1 \times G_2$. Moreover if both $\rho_1$ and $\rho_2$ are injective, then so is $\rho$. This construction is very similar to the direct sum of representations, except that two different groups intervene here. It seems also very related to the notion of product of groups.

Is there a name for such a construction, and has it been defined anywhere? Or am I completely missing something here?

toroloco
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You can do this. If you're trying to understand the representation theory of $G_1 \times G_2$ these representations aren't super helpful, because they are reducible (and there is another construction, the tensor product, that produces the irreducible representations of $G_1$ and $G_2$ out of the irreducible representations of $G_1$ and $G_2$). Nevertheless, the construction does produce a representation. Here's a previous question on this site that might be helpful: direct sum of representation of product groups

Chris Eagle
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