Let $G_1$, $G_2$ be finite groups. Let $V_1$ and $V_2$ be finite-dimensional complex representations of $G_1$ and $G_2$, respectively. Then their tensor product naturally becomes a representation which we call the outer tensor product, denoted $V_1 \boxtimes V_2$, via $(g_1, g_2) \cdot v_1 \otimes v_2 = {g_1 \cdot v_1} \otimes {g_2 \cdot v_2}$.
Now, let $\mathcal{G}$ be a set containing exactly one finite group from each isomorphism class of finite groups. Then, for each $G \in \mathcal{G}$, let $R(G)$ be the abelian group generated by the finite-dimensional complex representations of $G$, subject to the relations $[V] + [W] = [V + W]$.
Let $R = \bigoplus\limits_{G \in \mathcal{G}} R(G)$ be the direct sum of these abelian groups together with the outer tensor product as the ring multiplication. This turns $R$ into a $\mathcal{G}$-graded ring with unity. The irreducible representations form a basis of $R$ and are multipicatively closed.
We can also let $\tilde{R} = \prod\limits_{G \in \mathcal{G}} R(G)$ be the direct product of these abelian groups together with the outer tensor product as the ring multiplication. In this case, multiplication is well-defined since for a fixed finite group $G$ there are only finitely many pairs $(G_1, G_2)$ with $G_1 \times G_2 = G$. This turns $\tilde{R}$ into a ring with unity.
Now both $R$ and $\tilde{R}$ are somehow a "representation ring of all finite groups". There is an obvious inclusion $R \hookrightarrow \tilde{R}$. What can we say about the structure of $R$ and $\tilde{R}$?
Best Answer
The external tensor product is really not so interesting. If we combine the following two facts:
then it follows that $R$ is the following slightly funny structure: the tensor product of the tensor algebras of the $R(G_i)$ as $G_i$ ranges over the indecomposable finite groups. This is sort of a mix of a polynomial and noncommutative polynomial algebra; it is generated by generators corresponding to the irreducible representations of the indecomposables $G_i$, which do not commute with other irreducibles of the same group but do commute with irreducibles of other groups. $\widetilde{R}$ is the formal power series version of this. Edit: There is a subtlety that needs to be handled to make the multiplication well-defined, see the comments.
In particular this structure doesn't know anything about the ordinary tensor product of representations which is where the really interesting stuff is.