# Splitting fields of representations of generalized symmetric groups, 8

In this post, we give an example.

Let $G=C_3^8\, >\!\!\lhd \, S_8$ and let

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

where $\mu$ is a character of $C_3^8$ and $\rho$ is an irreducible representation of its stabilizer in $S_8$, $(S_8)_\mu$.

The real representations $\pi$ of $G$ are the ones for which

1. $\mu$ is represented by a character of the form

$(1,1,1,1,1,1,\omega,\omega^2) \ {\rm or}\ (1,1,...,1),$

and $\rho$ anything, or

2. $\mu$ is represented by a character of the form

$(1,1,1,1,\omega,\omega,\omega^2,\omega^2), \rho_1=(\pi_1,\pi_2,\pi_2)\in (S_4)^*\times (S_2)^*\times (S_2)^*,$

or

3. $\mu$ is represented by a character of the form

$(\omega,\omega,\omega,\omega,\omega^2,\omega^2,\omega^2,\omega^2), \rho_1=(\pi_2,\pi_2)\in (S_4)^*\times (S_4)^*,$

or

4. $\mu$ is represented by a character of the form

$(1,1,\omega,\omega,\omega,\omega^2,\omega^2,\omega^2), \rho_1=(\pi_1,\pi_2,\pi_2)\in (S_2)^*\times (S_3)^*\times (S_3)^*.$

The complex representations of $G$ are: the representations
whose characters have at least one complex value. Such representations $\pi = \theta_{\mu,\rho}$ are characterized by the fact that $(\mu,\rho)$ is inequivalent to $(\overline{\mu},\rho)$ under the obvious $S_8$-equivalence relation (which can be determined from the equivalence relation for representations in $G^*$).

The complex representations of $G$ are the remaining representations not included in the above list.

There are no quaternionic representations of $G$.

The claims above follow from the fact that a representation
$\theta_{\rho,\mu}$ is complex if and only if $\mu$ is not self-dual.

# Splitting fields of representations of generalized symmetric groups, 7

In this post, we discover which representations of the generalized symmetric group $G = S_n\ wr\ C_\ell = C_\ell^n\, >\!\!\lhd \, S_n$ can be realized over a given abelian extension of ${\mathbb{Q}}$.

Let $\theta_{\mu,\rho}\in G^*$ be the representation defined previously, where $\rho\in ((S_n)_\mu)^*$.

Let $K\subset {\mathbb{Q}}(\zeta_\ell)$ be a subfield, where $\zeta_\ell$ is a primitive $\ell^{th}$ root of unity. Assume $K$ contains the field generated by the values of the character of $\theta_{\mu,\rho}$. Assume $K/{\mathbb{Q}}$ is Galois and let $\Gamma_K=Gal({\mathbb{Q}}(\zeta_\ell)/K)$. Note if we regard $C_\ell$ as a subset of ${\mathbb{Q}}(\zeta_\ell)$ then there is an induced action of $\Gamma_K$ on $C_\ell$,

$\sigma:\mu \longmapsto \mu^\sigma, \ \ \ \ \ \ \ \ \ \mu\in (C_\ell)^*,\ \ \sigma\in \Gamma_K,$

where $\mu^\sigma(z)=\mu(\sigma^{-1}(z))$, $z\in C_\ell$. This action extends to an action on $(C_\ell^n)^*=(C_\ell^*)^n$.

Key Lemma:
In the notation above, $\theta_{\mu,\rho}\cong\theta_{\mu,\rho}^\sigma$ if and only if $\mu$ is equivalent to $\mu^\sigma$ under the action of $S_n$ on $(C_\ell^n)^*$.

Let

$n_\mu(\chi)=|\{i\ |\ 1\leq i\leq n,\ \mu_i=\chi\}|,$

where $\mu=(\mu_1,...,\mu_n)\in (C_\ell^n)^*$ and $\chi\in C_\ell^*$.

Theorem: The character of $\theta_{\mu,\rho}\in G^*$ has values in $K$ if and only if $n_\mu(\chi)=n_\mu(\chi^\sigma)$,
for all $\sigma\in \Gamma_K$ and all $\chi\in C_\ell^*$.

This theorem is proven in this paper.

We now determine the splitting field of any irreducible character of a generalized symmetric group.

Theorem: Let $\chi=tr(\theta_{\rho,\mu})$ be an irreducible character of $G=S_n\ wr\ C_\ell$. We have

$Gal({\mathbb{Q}}(\zeta_\ell)/{\mathbb{Q}}(\chi))= Stab_\Gamma(\chi).$

This theorem is also proven in this paper.

In the next post we shall give an example.

# Splitting fields of representations of generalized symmetric groups, 6

This post shall list some properties of the Schur index $m_F(G)$ in the case where $G = S_n\ wr\ C_\ell$ is a generalized symmetric group and $F$ is either the reals or rationals.

Let $\eta_k(z)=z^k$, for $z\in C_\ell$, $1\leq k\leq \ell$.

Theorem: Let $G = S_n\ wr\ C_\ell$. Let $\mu=(\eta_{e_1},...,\eta_{e_n})\in (C_\ell^n)^*$, for some $e_j\in \{0,...,\ell-1\}$, and let $\rho\in (S_n)_\mu^*$. Let
$\chi$ denotes the character of $\theta_{\mu,\rho}$.

1. Suppose that one of the following conditions holds:
1. $4|\ell$ and $\overline{e_1+...+e_n}$ divides $\overline{\ell/4}$ in ${\mathbb{Z}}/\ell {\mathbb{Z}}$, or
2. $(e_1+...+e_n,\ell)=1$,

Then $m_{\Bbb{Q}}(\chi)=1$.

2. Suppose that one of the following conditions holds:
1. $(n,\ell)=1$, $4|\ell$, and $(e_1+...+e_n)x\equiv \ell /4\ ({\rm mod}\ \ell)$ is not solvable, or
2. $(n,\ell)=1$ and $(e_1+...+e_n,\ell)>1$.

Then $m_{\mathbb{Q}}(\chi\eta_1)=1$.

This theorem is proven in this paper. Benard has shown that $m_{\mathbb{Q}}(\chi)=1$, for all $\chi$ as in the above theorem.

Since the Schur index over ${\mathbb{Q}}$ of any irreducible character $\chi$ of a generalized symmetric group $G$ is equal to $1$, each such character is associated to a representation $\pi$ all of whose matrix coefficients belong to the splitting field ${\mathbb{Q}}(\chi)$.

What is the splitting field ${\mathbb{Q}}(\chi)$, for $\chi\in G^*$?

This will be addressed in the next post.

# Splitting fields of representations of generalized symmetric groups, 5

It is a result of Benard (Schur indices and splitting fields of the unitary reflection groups, J. Algebra, 1976) that the Schur index over ${\mathbb{Q}}$ of any irreducible character of a generalized symmetric group is equal to $1$. This post recalls, for the sake of comparison with the literature, other results known about the Schur index in this case.

Suppose that $G$ is a finite group and $\pi \in G^*$ is an irreducible representation of $G$, $\pi :G\rightarrow Aut(V)$, for some complex vector space $V$. We say that $\pi$ may be realized over a subfield $F\subset {\mathbb{C}}$ if there is an $F$-vector space $V_0$ and an action of $G$ on $V_0$ such that $V$ and ${\mathbb{C}}\otimes V_0$ are equivalent representations of $G$, where $G$ acts on ${\mathbb{C}}\otimes V_0$ by “extending scalars” in $V_0$ from $F$ to ${\mathbb{C}}$. Such a representation is called an $F$-representation. In other words, $\pi$ is an $F$-representation provided it is equivalent to a representation which can be written down explicitly using matrices with entries in $F$.

Suppose that the character $\chi$ of $\pi$ has the property that

$\chi(g)\in F, \ \ \ \ \ \ \forall g\in G,$

for some subfield $F\subset {\mathbb{C}}$ independent of $g$. It is unfortunately true that, in general, $\pi$ is not necessarily an $F$-representation. However, what is remarkable is that, for some $m\geq 1$, there are $m$ representations, $\pi_1,...,\pi_m$, all equivalent to $\pi$, such that $\pi_1\oplus ...\oplus \pi_m$ is an $F$-representation. The precise theorem is the following remarkable fact.

Theorem: (Schur) Let $\chi$ be an irreducible character and let $F$ be any field containing the values of $\chi$. There is an integer $m \geq 1$ such that $m\chi$ is the character of an $F$-representation.

The smallest $m\geq 1$ in the above theorem is called the Schur index and denoted $m_F(\chi)$.

Next, we introduce some notation:

1. let ${\mathbb{R}}(\pi) = {\mathbb{R}}(\chi)$ denote the extension field of ${\mathbb{R}}$ obtained by adjoining all the values of $\chi(g)$\ ($g\in G$), where $\chi$ is the character of $\pi$,
2. let $\nu(\pi) = \nu(\chi)$ denote the Frobenius-Schur indicator of $\pi$ (so $\nu(\pi)= {1\over |G|}\sum_{g\in G} \chi(g^2)$),
3. let $m_{\mathbb{R}}(\pi) = m_{\mathbb{R}}(\chi)$ denote the Schur multiplier of $\pi$ (by definition, the smallest integer $m\geq 1$ such that $m\chi$ can be realized over ${\mathbb{R}}$ (this integer exists, by the above-mentioned theorem of Schur).

The following result shows how the Schur index behaves under induction (see Proposition 14.1.8 in G. Karpilovsky,
Group representations, vol. 3, 1994).

Proposition: Let $\chi$ be an irreducible character of $G$ and let $\psi$ denote an irreducible character of a subgroup $H$ of $G$. If $= 1$ then $m_{\Bbb{R}}(\chi)$ divides $m_{\Bbb{R}}(\psi)$.

A future post shall list some properties of the Schur index in the case where $G$ is a generalized symmetric group and $F$ is either the reals or rationals.

# Splitting fields of representations of generalized symmetric groups, 4

This post if an aside on cyclotomic fields and Tchebysheff polynomials. Though it seems certain this material is known, I know of no reference.

Let $n$ denote a positive integer divisible by $4$, let $r=\cos(2\pi/n)$, $s=\sin(2\pi/n)$, and let $d=n/4$. If

$T_1(x)=x,\ \ T_2(x)=2x^2-1,\ \ T_3(x)=4x^3-3x,\ \ T_4(x)=8x^4-8x^2+1,\ \ ...,$

denote the Tchebysheff polynomials (of the 1st kind), defined by

$\cos(n\theta)=T_n(\cos(\theta)),$

then $T_d(r)=0.$

Let $\zeta_n=exp(2\pi i/n)$ and let $F_n={\mathbb{Q}}(\zeta_n)$ denote the cyclotomic field of degree $\phi(n)$ over ${\mathbb{Q}}$. If $\sigma_j\in Gal(F_n/{\Bbb{Q}})$ is defined by $\sigma_j(\zeta_n)=\zeta_n^j$ then

$Gal(F_n/{\Bbb{Q}})\cong ({\Bbb{Z}}/n{\Bbb{Z}})^\times,$

where $\sigma_j\longmapsto j$.

Lemma: Assume $n$ is divisible by $4$.

1. ${\mathbb{Q}}(r)$ is the maximal real subfield of $F_n$, Galois over ${\mathbb{Q}}$ with

$Gal(F_n/{\Bbb{Q}}(r))=\{1,\tau\},$

where $\tau$ denotes complex conjugation. Under the canonical isomorphism

$Gal(F_n/{\Bbb{Q}})\cong ({\Bbb{Z}}/n{\Bbb{Z}})^\times,$

we have

$Gal({\Bbb{Q}}(r)/{\Bbb{Q}})\cong ({\Bbb{Z}}/n{\Bbb{Z}})^\times/\{\pm 1\}.$

2. If $n$ is divisible by $8$ then $r$ and $s$ are conjugate roots of $T_d$. In particular, $s\in {\mathbb{Q}}(r)$ and $T_d(s)=0$.

3. We have $\sigma_j(r)=T_j(r)$.
4. If $n\geq 4$ is a power of $2$ then $T_d$ is the minimal polynomial of ${\mathbb{Q}}(r)$. Furthermore, in this case

$\cos(\pi/4)=\sqrt{2}/2,\ \ \cos(\pi/8)=\sqrt{2+\sqrt{2}}/2,\ \ \cos(\pi/16)=\sqrt{2+\sqrt{2+\sqrt{2}}}/2,\ \ ... .$

# 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.