# “Tensor complement” of representations of finite groups: reloaded

Let $$V$$ be a finite dimensional simple $$G$$-representation (over $$\mathbb{C}$$) for a finite group $$G$$. Let $$R$$ be the regular representation of $$G$$.

Is there a $$G$$-representation $$W$$ and $$k\geq 1$$ such that $$V \otimes W \cong R^{\otimes k}$$?

Note: if we demand $$k=1$$, then this is false (see here).

The character of $$R$$ takes the identity to $$|G|$$ and all other elements to zero. Therefore the character of $$R^{\otimes k}$$ takes the identity to $$|G|^k$$ and all other elements to zero.
Let $$\chi$$ be the character of $$G$$, and let $$\chi(e)=m$$. Then $$m\mid n$$ where $$n=|G|$$, since $$\chi$$ is irreducible. If we let $$W_1$$ be the direct sum of $$n/m$$ copies of the trivial representation, then $$V\otimes W_1$$ has dimension $$n$$. Let $$W=R_1\otimes R$$. Then $$V\otimes W\cong (V\otimes W_1)\otimes R$$ has the same character as $$R^{\otimes 2}$$, so is isomorphic to it.