Finite Groups – When is $M_n(\mathbb{Q}(\zeta))$ a Factor of $\mathbb{Q}[G]$?

finite-groupsgr.group-theoryrt.representation-theory

I am cross-posting this question from my MSE post here, in case someone here can answer it.

For a finite group $G$, the rational group ring $\mathbb{Q}[G]$ has a Wedderburn decomposition:
$$
\mathbb{Q}[G] \cong \prod_{i=1}^k M_{n_i}(D_i),
$$
where $D_i$ is a division algebra whose center is a number field, and $M_{n_i}(D_i)$ is the ring of $n_i \times n_i$ matrices over $D_i$. I will refer to the factors of this product as the simple factors of $\mathbb{Q}[G]$. Each simple factor corresponds to an irreducible $\mathbb{Q}$-representation of $G$.

My question is: given $n$ and a primitive $d$th root of unity $\zeta_d$, for which finite groups $G$ does $M_n(\mathbb{Q}(\zeta_d))$ appear as a simple factor of $\mathbb{Q}[G]$ corresponding to a faithful representation of $G$?

It is well known that if $G$ is cyclic of order $d$, then the unique faithful irreducible $\mathbb{Q}$-representation of $G$ has $\mathbb{Q}(\zeta_d)$ as its corresponding simple factor.
Moreover, if $G$ is a Heisenberg group over $\mathbb{Z}/p\mathbb{Z}$, then it has a unique faithful irreducible representation over $\mathbb{Q}$ and the corresponding simple factor is $M_p(\mathbb{Q}(\zeta_p))$; Kenta S sketched a proof of this in the MSE thread.

What other examples are there?

Best Answer

In general, I cannot say very much. Let $K = \mathbb Q(\zeta_d)$. Suppose $V_{\mathbb Q}$ is a simple $\mathbb QG$-module corresponding to a simple factor $M_n(K)$. Then $V_{\mathbb Q}$ comes from restriction of scalars along $K/\mathbb Q$ of a $KG$-module $V$ which is absolutely irreducible. Then $V$ is faithful if and only if $V_\mathbb Q$ is faithful. So it suffices to understand when $KG$ has a faithful absolutely irreducible module. This decomposes into two parts:

  1. When does $G$ have a faithful irreducible representation with character valued in $K$?

  2. When is a character from 1. defined over $K$?

The question of when a group has a complex faithful irreducible representation is discussed in the following MO thread: Which finite groups have faithful complex irreducible representations?. The second point is a question of Schur indices and I cannot say much about it in general.

In the case of a (finite) $p$-group, then there is a neat criterion. A $p$-group $P$ has a faithful complex irreducible representation if and only if $Z(P)$ is cyclic [1]. Furthermore Schur indices vanish for $p$-groups when $p > 2$. The second point was used by Ford to prove the following theorem:

Theorem [2]: Each (nontrivial) irreducible rational representation of a finite $p$-group is induced from the irreducible faithful rational representation of degree $p-1$ on a section of order $p$.

That is, each simple factor of $\mathbb QP$ for $p > 2$ is of the form $M_n(\mathbb Q(\zeta_p))$.

Corollary: For a $p$-group $P$ with $p > 2$, there is a simple factor $M_n(\mathbb Q (\zeta_d))$ of $\mathbb QP$ corresponding to a faithful representation if and only if $P$ has cyclic center and $d = p$.

So, the Heisenberg example is a general phenomenon for odd $p$-groups.

References:

[1] Isaacs, Character Theory, Theorem 2.32

[2] Ford, Charles E. "Characters of p-groups." In Proc. Am. Math. Soc, vol. 101, no. 4, pp. 595-600. 1987.

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