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^*\},

by

(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:

  1. For each \mu \in [A^*] and \rho\in H_\mu^*, as above, then \theta_{\mu,\rho} is an irreducible representation of G.
  2. 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.
  3. 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.

In the next post, we will examine the special case A=C_\ell^n and H=S_n.

Leave a Reply

Please log in using one of these methods to post your comment:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s