Suppose $G$ is a finite group and $V$ is a finite-dimensional representation of $G$ over a field $k$. I'd like to write $V$ as a tensor product $V\_1 \otimes V\_2 \otimes \dots V\_n$ satisfying


(1) Each $V\_i$ is a representation of $G$.


(2) No $V\_i$ can be written as a tensor product of two or more $G$-representations of dimension > 1.


I'd like to call it a "prime factorization" of $V$, and each $V\_i$, a prime representation of $V$?


Can this always be done? Is this prime factorization unique up to tensoring with one-dimensional representations (the units) and isomorphisms? Given $G$ and $k$ how can I write down all the prime representations of $G$ over $k$? What is a good reference for this stuff?


[Edited after R.'s comment.]



Prime factorization of a group representation
These factorizations are not unique, and there is already a counterexample for the smallest nontrivial group $G = C\_2$. The trivial representation has character $(1, 1)$ (where the first entry is the value of the character on the identity and the second entry is the value of the character on $-1$), and the sign representation has character $(1, -1)$, so representations can have character $(n + m, n - m)$ where $n, m$ are nonnegative integers. dhy in the comments gives the counterexample


$$(4, 0) = (2, 2) \times (2, 0) = (2, 0) \times (2, 0)$$


showing that we already don't have uniqueness here (any representation of prime dimension is prime). We can even arrange for the character values to be nonzero, for example


$$(35, 15) = (5, 5) \times (7, 3) = (5, 3) \times (7, 5).$$


Overall this is not a very well-behaved notion, I think; there are infinitely many representations of prime dimension given by taking direct sums of irreducibles whose dimension happens to add up to a prime. This is not a very natural condition and I don't think there's anything useful to say about it.


