Rational-representation group

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Definition

A rational-representation group is a finite group satisfying the following equivalent properties:

1. The field of rational numbers $\mathbb{Q}$ is a splitting field, i.e., every irreducible representation in characteristic zero is realizable over the rational numbers.
2. Every irreducible representation in characteristic zero can be realized over $\mathbb{Z}$.

Equivalence of definitions

Definitions (1) and (2) are equivalent because linear representation is realizable over principal ideal domain iff it is realizable over field of fractions.

Examples

Here are some small examples, with relevant contextual information.

Group	Order	List of degrees of irreducible representations	List of entries of matrices arising across all irreducible representations when chosen with simplest entries
trivial group	1	1	$\{ 1 \}$
cyclic group:Z2	2	1,1	$\{ 1, -1 \}$
Klein four-group	4	1,1,1,1,	$\{ 1,-1 \}$
symmetric group:S3	6	1,1,2	$\{ 1,0,-1 \}$
elementary abelian group:E8	8	1,1,1,1,1,1,1,1	$\{ 1, -1 \}$
dihedral group:D8	8	1,1,1,1,2	$\{ 1,0,-1 \}$
direct product of S3 and Z2	12	1,1,1,1,2,2	$\{ 1,0,-1 \}$
elementary abelian group:E16	16	1 (16 times)	$\{ 1,-1 \}$
direct product of D8 and Z2	16	1,1,1,1,1,1,1,1,2,2	$\{ 1,0,-1 \}$
symmetric group:S4	24	1,1,2,3,3	$\{ 1,0,-1 \}$
direct product of S3 and V4	24	1 (8 times), 2 (4 times)	$\{ 1,0,-1 \}$

Facts

• Symmetric groups on finite sets are rational-representation groups. For full proof, refer: Symmetric groups are rational-representation
• Dihedral group:D8 is a rational-representation group.
• Trivial group, cyclic group:Z2, and dihedral group:D8 are the only three groups with the property that all their irreducible representations can be written over the rationals as orthogonal matrices. For full proof, refer: Classification of finite groups for which every irreducible representation can be realized using orthogonal matrices over the rational numbers

Relation with other properties

Weaker properties

• Rational group: A rational group is a finite group such that all its characters are rational-valued (hence integer-valued). The quaternion group is an example of a rational group that is not a rational-representation group.
• Ambivalent group: A finite group in which every character is real-valued.