In Section 3.2 of Linear Representations of Finite Groups by Serre, the author gives a formula to compute the characters of product of finite groups (Abelian or non-Abelian).

First Serre defines the product of two groups.

Let $G_1$ and $G_2$ be two groups, and let $G_1 \times G_2$ be their product, that is, the set of pairs $\left(s_1, s_2\right)$, with $s_1 \in G_1$ and $s_2 \in G_2$. Then,

$$ \left(s_1, s_2\right) \cdot \left(t_1, t_2\right) = \left(s_1 t_1, s_2 t_2\right) $$

Then, Serre introduced the representation of the product as follows.

Now let $\rho^1 : G_1 \to GL\left(V_1\right)$ and $\rho^2 : G_2 \to GL\left(V_2\right)$ be linear representations of $G_1$ and $G_2$ respectively. We define a linear representation $\rho^1 \otimes rho^2$ of $G_1 \times G_2$ into $V_1 \otimes V_2$ by setting

$$\left(\rho^1 \otimes \rho^2\right)\left(s_1, s_2\right) = \rho^1 \left(s_1\right) \otimes \rho^2 \left(s_2\right)$$

Finally, comes the formula for computing the characters of the product.

$$\chi \left(s_1, s_2\right) = \chi_1 \left(s_1\right) \cdot \chi_2 \left(s_2\right)$$

In the same section, Serre introduces Theorem 10 which is as follows.

Theorem 10: (i) If $\rho^1$ and $\rho^2$ are irreducible, $\rho^1 \otimes \rho^2$ is an irreducible representation of $G_1 \times G_2$.

(ii) Each irreducible representation of $G_1 \times G_2$ is isomorphic to a representation $\rho^1 \otimes \rho^2$, where $rho^i$ is an irreducible representation of $G_i (i = 1, 2)$.

My question: Do we have similar results for semidirect and wreath products of non-Abelian finite groups?

My effort: I have found in Section 8.2 of Serre how to compute the characters when one of the group is Abelian. I have no clue how to extend the idea for non-Abelian group. In Representations of semidirect products, Akhil Mathew defines the irreps of semidirect products but it is not clear to me how to compute the character of the product.

  • 1
    $\begingroup$ Omar I was looking for the same thing while back and I didn't find something concrete unfortunately. At Etingof's Introduction to representation theory there is a section for representations of semidirect products. Though isn't something fancy. $\endgroup$ – user321268 Oct 21 '16 at 12:04
  • $\begingroup$ @mayer_vietoris, let me check the reference you mentioned. $\endgroup$ – Omar Shehab Oct 22 '16 at 22:25
  • $\begingroup$ @mayer_vietoris, did you check this (jlms.oxfordjournals.org/content/s2-13/2/281.extract)? $\endgroup$ – Omar Shehab Oct 23 '16 at 3:20

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