Splitting fields of representations of generalized symmetric groups, 3

The representations of a semi-direct product of a group $H$ by an abelian group $A$ , written $G=A\, >\!\!\lhd \, H$ (so $A$ is normal in $G$ ) can be described explicitly in terms of the representations of $A$ and $H$ . The purpose of this post is to explain how this is done.

Again, a good reference for all this is Serre’s well-known book, Linear representations of finite groups.

Let $f$ be a class function on $H$. Extend $f$ to $G$ trivially as follows:

$f^0(g)= \left\{ \begin{array}{cc} f(g),&g\in H,\\ 0, & g\notin H, \end{array} \right.$

for all $g\in G$ . This is not a class function on $G$ in general. To remedy this, we “average over $G$ ” using conjugation: Define the function $f^G=Ind_H^G(f)$ induced by $f$ to be

$Ind_H^G(f)(g)={1\over |H|}\sum_{x\in G} f^0(x^{-1}gx)=\sum_{x\in G/H}f^0(x^{-1}gx).$

This is referred to as the Frobenius formula.

Since $A$ is normal in $G$ , $G$ acts on the vector space of formal complex linear combinations of elements of $A^*$ (=the characters of $A$ ),

$V={\mathbb{C}}[A^*]=span\{\mu\ |\ \mu\in A^*\},$

$(g\mu)(a)=\mu(g^{-1}ag),\ \ \ \ \forall g\in G,\ a\in A,\ \mu\in A^*.$

We may restrict this action to $H$ , giving us a homomorphism $\phi^*:H\rightarrow S_{A^*}$ , where $S_{A^*}$ denotes the symmetric group of all permutations of the set $A^*$ . This restricted action is an equivalence relation on $A^*$ which we refer to below as the $H$ -equivalence relation}. Let $[A^*]$ denote the set of equivalence classes of this equivalence relation. If $\mu,\mu'$ belong to the same equivalence class then we write

$\mu'\sim \mu$

(or $\mu'\sim_H\mu$ if there is any possible ambiguity). When there is no harm, we identify each element of $[A^*]$ with a character of $A$ .

Suppose that $H$ acts on $A$ by means of the automorphism given by a homomorphism $\phi:H\rightarrow S_{A}$ , where $S_{A}$ denotes the symmetric group of all permutations of the set $A$ . In this case, two characters $\tau,\tau'\in A^*$ are equivalent if there is an element $h\in H$ such that, for all $a\in A$ , we have $\tau'(a)=\tau(\phi(h)(a))$ .

For each $\mu\in [A^*]$ , let

$H_{\mu}=\{h\in H\ |\ h\mu = \mu\}.$

This group is called the stabilizer of $\mu$ in $H$ . Let

$G_{\mu}=A\, >\!\!\lhd \, H_{\mu},$

for each $\mu\in [A^*]$ . There is a natural projection map

$p_{\mu}:G_{\mu}\rightarrow H_{\mu}$

given by $ah\longmapsto h$ , i.e., by $p_\mu(ah)=a$ .

Extend each character $\mu\in [A^*]$ from $H_{\mu}$ to $G_{\mu}$ trivially by defining

$\mu(ah)=\mu(a),$

for all $a\in A$ and $h\in H_{\mu}$ . This defines a character $\mu\in G^*_{\mu}$ . For each $\rho\in H_{\mu}^*$ , say $\rho:H_{\mu} \rightarrow Aut(V)$ , let $\tilde{\rho}\in G_{\mu}^*$ denote the representation of $G_{\mu}$ obtained by pulling back $\rho$ via the projection $p_\mu:G_{\mu}\rightarrow H_{\mu}$ , i.e., define

$\tilde{\rho}=\rho\circ p_{\mu}.$

For each $\mu \in [A^*]$ and $\rho\in H_\mu^*$ as above, let

$\theta_{\mu,\rho}=Ind_{G_\mu}^G(\mu\cdot \tilde{\rho}).$

Finally, we can completely describe all the irreducible representations of $G=A\, >\!\!\lhd \, H$ . (This is Proposition 25 in chapter 8 of Serre’s book.)

Theorem:

For each $\mu \in [A^*]$ and $\rho\in H_\mu^*$ , as above, then $\theta_{\mu,\rho}$ is an irreducible representation of $G$ .
Suppose $\mu_1,\mu_2 \in [A^*]$ , $\rho_1\in H_{\mu_1}^*$ , $\rho_2\in H_{\mu_2}^*$ . If $\theta_{\mu_1,\rho_1}\cong \theta_{\mu_2,\rho_2}$ then $\mu_1\sim \mu_2$ and $\rho_1\cong \rho_2$ .
If $\pi$ is an irreducible representation of $G$ then $\pi\cong \theta_{\mu,\rho}$ , for some $\mu \in [A^*]$ and $\rho\in H_{\mu}^*$ as above.