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, 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.