# Schur index of irreducible character

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## Contents

## Definition

### Direct definition

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

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

The Schur index of a character is often denoted .

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 -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 . The representation cannot be realized over (this follows from the indicator theorem) but it can be realized in any quadratic extension of the form for .
- 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.