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.