Schur index of irreducible character

Definition

Direct definition

Suppose $G$ is a finite group, $K$ is a splitting field for $G$, and $\chi$ is the character of an irreducible linear representation $\varphi$ of $G$ over $K$. Suppose $k$ is the subfield of $K$ generated by the character values $\chi (g),g\in G$. The Schur index of $\chi$ (also termed the Schur index of $\varphi$) is defined in the following equivalent ways:

1. It is the smallest positive integer $m$ such that there exists a degree $m$ extension $L$ of $k$ such that $\varphi$ can be realized over $L$, i.e., we can change basis so that all the matrix entries are from $L$. Note that it is not necessary that $L$ be a subfield of $K$, but rather we need to work within a suitable larger field that contains both $L$ and $K$ to perform the necessary conjugation.
2. It is the multiplicity of $\varphi$ in any irreducible linear representation $\alpha$ of $G$ over $k$ that has $\varphi$ as one of its irreducible constituents over $K$.

The Schur index of a character $\chi$ is often denoted $m(\chi )$.

Note that if the representation can be realized over the field generated by the character values for that representation, the Schur index is one.

Definition in terms of division rings

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.

Facts

• Schur index of irreducible character need not equal degree of extension of minimal field realizing the character over field generated by character values
• Schur index of irreducible character is one in any prime characteristic
• Schur index divides degree of irreducible representation
• Schur index of irreducible character in characteristic zero divides exponent
• Square of Schur index of irreducible character in characteristic zero divides order
• Odd-order p-group implies every irreducible representation has Schur index one

Examples

From the facts above, it is clear that to get an example of an irreducible character/representation with Schur index greater than 1, we should not look at odd-order $p$-groups. Here are the smallest examples:

• The smallest example with Schur index two is faithful irreducible representation of quaternion group. The field generated by character values is $\mathbb {Q}$. The representation cannot be realized over $\mathbb {Q}$ (this follows from the indicator theorem) but it can be realized in any quadratic extension of the form $\mathbb {Q} ({\sqrt {-m^{2}-1}})$ for $m\in \mathbb {Q}$.
• The smallest example with Schur index three is a representation of the nontrivial semidirect product of Z7 and Z9 (order 63). Note that we cannot find any examples of Schur index three using 3-groups, because odd-order p-group implies every irreducible representation has Schur index one.