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.