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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.

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    $\begingroup$ I suppose you are implicitly looking at $\mathbb{C}$ representations, and maybe you should be explicit. I just say that because in finite characteristic this can happen more readily: eg, the two different $2$-diml representations of $A_5=SL(2,4)$ in characteristic $2$ tensor to give the $4$-diml irreducible; and more generally the irreducible representations of $SL(2,2^n)$ tensor together "nicely". $\endgroup$ Jan 9 at 15:00
  • $\begingroup$ that's an interesting example, but you're right that I'm really interested in representations over $ \mathbb{C} $, I'll mention that $\endgroup$ Jan 9 at 18:19

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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.

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  • $\begingroup$ Ah! The first paper is exactly the sort of thing I'm interested in. I'm a big fan of Tiep's work. I'd already seen all the cases in Theorem A , and I'd also seen the $ (6.A_7,6) $ case from Theorem B. But I had been thinking that WLOG at least one character should be faithful, but they point out that the intersection of the two kernels could be disjoint without either being faithful. That' very interesting. I'd like to think more about their examples like $ (4^2.PSL_3(4),8) $ and especially $ (2^2×3.PSL_3(4), 6) $ and $ ((3^2 × 2).PSU_4(3),6) ) $ $\endgroup$ Jan 13 at 15:02
  • $\begingroup$ I am glad you liked the reference, and I agree, both Navarro or Tiep are very interesting authors/mathemeticians! The faithfulness also plays a role in the paper of Lewis and Hendrixson. $\endgroup$ Jan 13 at 15:45
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    $\begingroup$ another reference that covers quasisimple groups, like the Navarro Tiep paper given as the first reference, is this other Tiep paper semanticscholar.org/paper/… even though it is much older it is in some ways a more comprehensive review of irreducible products of characters. Also turns out my $ A_9 $ example from the original question is of the general pattern for $ A_{m^2} $ for all $ m \geq 3 $! This is covered in "Irreducible products of characters in $ A_n $" by Ilan Zisser. $\endgroup$ Feb 5 at 1:39

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