Minimal ring of realization of irreducible representations

Definition

In characteristic zero

Suppose $G$ is a finite group and $R$ is an integral domain of characteristic zero, i.e., it contains the ring of integers as a subring. We say that $R$ is a minimal ring of realization of irreducible representations if all irreducible representations of $G$ over some splitting field containing $R$ can be realized with matrix entries all from $R$ and such that no subring of $R$ has this property.

Facts

• Linear representation is realizable over principal ideal domain iff it is realizable over field of fractions: In particular, if $R$ is a principal ideal domain, then any representation that can be realized over the field of fractions can also be realized over $R$. This fact allows us to compute some minimal rings of realization.
• Minimal ring of realization of irreducible representations need not be unique
• Minimal rings of realization of irreducible representations for the same group may have different degrees as extensions of the ring of integers