I am looking an example of an irreducible complex character $\chi$ of a finite group $G$ such that $\chi\otimes \chi$ decomposes as symmetric square and alternating square of $\chi$, and these two components are irreducible also.
Question 1. From the character table, I guess, there is an irreducible character of $SL_2(3)$ over $\mathbb{C}$, of degree $2$ whose values are all in $\mathbb{Z}$ (see here) and its tensor square decomposes as sum of symmetric square and alternating square, with both components irreducible; is this correct?
Question 2. Are there any other examples of finite groups in which symmetric and alternating square of an irreducible complex character are both irreducible?
Question 3. In general, is it investigated, when symmetric square of a complex irreducible character is irreducible or when alternating square of an irreducible character is irreducible?