# Splitting fields of representations of generalized symmetric groups, 4

First a technical definition.

Let $A=C_\ell^n$. Let $\eta_k(z)=z^k$, for $z\in C_\ell$ and $1\leq k\leq \ell-1$. For $\eta\in C_\ell^*$, let $\mu\otimes \eta =(\mu_1\eta,\mu_2\eta,...,\mu_n\eta)$ where $\mu=(\mu_1,\mu_2,...,\mu_n)$. This defines an action of $C_\ell^*$ on $A^*$ and hence on the set of equivalence classes of $G$, $G^*$. We call two representations $\theta_{\mu,\rho}$, $\theta_{\mu',\rho'}$ $C_\ell^*$-equivalent, and write

$\theta_{\mu,\rho}\sim_\ell \theta_{\mu',\rho'},$

if $\rho=\rho'$ and $\mu'=\mu\otimes \eta$ for some $\eta\in C_\ell^*$. Similarly, we call two characters $\mu$, $\mu'$ of $C_\ell^n$ $C_\ell^*$-equivalent, and write

$\mu\sim_\ell \mu',$

if $\mu'=\mu\otimes \eta$ for some $\eta\in C_\ell^*$.

For example, Let $\ell =9$, $n=3$ and $\mu=(\eta_2,\eta_5,\eta_8)$. Then $\mu\sim \mu\otimes\eta_3$.

Let $\theta_{\mu,\rho}$ be as in the previous post. Note that

$\theta_{\mu\otimes \eta,\rho} = \theta_{\mu,\rho}\otimes \eta ,$

for $\eta\in C_\ell^*$. Therefore, the matrix representations of two $C_\ell^*$-equivalent representations differ only
by a character.

Let $G=C_\ell^n\, >\!\!\lhd \, S_n$.

The results in the above section tells us how to construct all the irreducible representations of $G$. We must

1. write down all the characters (i.e., 1-dimensional representations) of $A=C_\ell^n$,
2. describe the action of $S_n$ on $A^*$,
3. for each $\mu\in [A^*]$, compute the stabilizer $(S_n)_{\mu}$,
4. describe all irreducible representations of each $(S_n)_{\mu}$,
5. write down the formula for the character of $\theta_{\mu,\rho}$.

Write $\mu\in [A^*]$ as $\mu=(\mu_1,...,\mu_n)$, where each component is a character of the cyclic group $C_\ell$, $\mu_j\in C_\ell^*$. Let $\mu'_1,...,\mu'_r$ denote all the distinct characters which occur in $\mu$, so

$\{\mu'_1,...,\mu'_r\}=\{ \mu_1,...,\mu_n\}.$

Let $n_1$ denote the number of $\mu'_1$‘s in $\mu$, $n_2$ denote the number of $\mu'_2$‘s in $\mu$, …, $n_r$ denote the number of $\mu'_r$‘s in $\mu$. Then $n=n_1+...+n_r$. Call this the partition associated to $\mu$.

If two characters $\mu=(\mu_1,...,\mu_n)$, $\mu'=(\mu'_1,...,\mu'_n)$ belong to the same class in $[(C_\ell^n)^*]$, under the $S_n$-equivalence relation, then their associated partitions are equal.

The Frobenius formula for the character of an induced representation gives the following character formula. Let $\chi$ denote the character of $\theta_{\mu,\rho}$. Then

$\chi(\vec{v},p)=\sum_{g\in S_n/(S_n)_\mu} \chi^o_\rho(gpg^{-1})\mu^g(\vec{v}),$

for all $\vec{v}\in C_\ell^n$ and $p\in S_n$. In particular, if $p=1$ then

$\chi(\vec{v},1)=({\rm dim}\ \rho)\sum_{g\in S_n/(S_n)_\mu} \mu^g(\vec{v}).$

# 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 $H=S_n$.

# Splitting fields of representations of generalized symmetric groups, 2

In general, there are three types of (complex) representations of a finite group $G$. (A good reference for all this is Serre’s well-known book, Linear representations of finite groups.)

Let $\rho:H\rightarrow Aut(W)$ be an $n$-dimensional irreducible representation of a finite group $G$ on a complex vector space $W$. Let $\chi$ denote the character of $\rho$.

Exactly one of the following possibilities must hold:

• One of the values of the character $\chi$ is not real. Such representations will be called complex (or type 1 or unitary).
• All the values of $\chi$ are real and $\rho$ is realizable by a representation over a real vector space. Such representations will be called real (or type 2 or orthogonal).
• All the values of $\chi$ are real but $\rho$ is not realizable by a representation over a real vector space. Such representations will be called quaternionic (or type 3 or symplectic).

Proposition (Frobenius-Schur): Let $\rho:H\rightarrow Aut(W)$ be an irreducible representation of a finite group $G$ on a complex vector space $W$ with character $\chi$. Then

${1\over |G|} \sum_{g\in G}\chi(g^2)= \left\{ \begin{array}{cc} 0,&\rho\ {\rm complex},\\ 1,&\rho\ {\rm real},\\ -1,&\rho\ {\rm quaternionic}. \end{array} \right.$

This quantity is sometimes called the Frobenius-Schur indicator of $\rho$.

It can be shown that if $\rho$ $\rho'\cong \rho$ are equivalent representations then $\rho$ and $\rho'$ have the same type.

In the next post we will examine the types that the irreducible representations of semi-direct product $G=C_\ell^n\, >\!\!\lhd \, S_n$ fall into.

# Splitting fields of representations of generalized symmetric groups, 1

The main result of this series of blog posts (originally written 1998), which is expository, is to determine (in a sense made precise later) the splitting field of any irreducible character of a generalized symmetric group. This was basically solved by M. Benard in a 1976 J. of Algebra paper. We use one of his results to make the splitting field explicit.

Notation and definitions

Let $C_\ell$ denote the cyclic group of order $\ell\geq 1$, let $S_n$ denote the symmetric group of degree $n\geq 1$, and let $G$ denote the semi-direct product $G=C_\ell^n\, >\!\!\lhd \, S_n$. We think of this as the set of pairs $(v,p)$, with

• $v=(v_1,...,v_n)$, where each $v_i\in C_\ell=\{0,1,...,\ell-1\}$,
• $p\in S_n$,
• $S_n$ acts on $C_\ell^n$ by $p(v)=(v_{p(1)},v_{p(2)},...,v_{p(n)})$,
• multiplication given by $(v,p)*(v',p')=(v+p(v'),pp'),\ \ \ \ \ (v,p),(v',p')\in G.$

A group of the form $C_\ell^n\, >\!\!\lhd \, S_n$, also written $S_n \ {\rm wr}\ C_\ell$ (here wr denotes the wreath product), will be called a generalized symmetric group.

We may identify each element of $C_\ell^n\, >\!\!\lhd \, S_n$ with an $n\times n$ monomial matrix (a matrix with exactly one non-zero entry in each row and column) with entries in $C_\ell$.

Our main goal will be to investigate the following question: What is the smallest extension of ${\mathbb{Q}}$ required to realize (using matrices) a given irreducible character of a generalized symmetric group?