All representations are considered over the complex numbers.
Let $ G $ be a finite group. Then Any 1-dimensional character $\otimes$ irreducible character is irreducible .
But what if we have two irreducible characters both of degree greater than 1? How often will it be the case that the tensor product of two irreps of degree $ \geq 2 $ will still be irreducible?
For some groups there are no irrep triples $ \chi_1 \chi_2=\chi_3 $ with $ dim(\chi_1), dim(\chi_2) \geq 2 $. For example, $ A_5 $ only has irreps of degrees $ 1,3,3,4,5 $ and $ 3^2>5 $ so no such triple exist.
But triples like this certainly aren't impossible. A generic construction of some example triples is given in Representations irreducible with respect to the tensor product . However those examples take a tensor product of two representations, neither of which are faithful.
For an example using irreps which are not all faithless, take $ G=2.A_5 $ then we have
examples like $ \pi_{2,1} \otimes \pi_{2,2}= \pi_{4,1} $ and $ \pi_{2,1} \otimes \pi_{3,2}= \pi_{6} $ and $ \pi_{2,2} \otimes \pi_{3,1} = \pi_{6} $ where the first index denote the dimension of the irrep, and if there are multiple irreps of the same dimension then the second index represents the order it is listed by GAP in the command CharacterTable("2.A5"). For example $ \pi_{4,1} $ denotes the first 4d irrep listed by GAP (the non-faithful one).
Can we say anything about which finite groups admit irrep triples like this? For example $ A_9 $ and the monster group have such triples.
Certainly if $ G $ has character triples like this, then any group with $ G $ as a quotient also has character triples like this, so the question is most interesting when at least one of the characters in the triple is faithful.
Best Answer
This is a great question, but difficult to answer. See for example for special cases: the paper of Gabriel Navarro and Pham Huu Tiep, On irreducible products of characters (2021), or this one by Mark Lewis and Lisa Hendrixson, Products of Irreducible Characters Having Complex-Valued Constituents (2017). Hope this helps.