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

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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.