Representation Theory – Irreducibility of a Tensor Product of Two Irreducible Representations

finite-groupsrepresentation-theory

This is a follow-up question to Chaturvedi's question and Rickard's reply
— see Representations irreducible with respect to the tensor product

Question: Suppose $U$ and $V$ are two irreducible complex representations of a finite group $G$ such that the tensor product $U \otimes V$
is also irreducible. Must it be the case that either $U$
or $V$ (or both) are one dimensional ?

Thanks, Ines

Best Answer

How about this? Take groups $H_1$ and $H_2$ with irreps $U_1$ and $U_2$. Let $G=H_1\times H_2$. Regard $U_i$ as a $G$-representation by pulling back the $H_i$-action to $G$ via the projection $G\to H_i$. I reckon then $U_1$, $U_2$ and $U_1\otimes U_2$ will all be irreps of $G$.

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