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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).

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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.

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