# Minimal ring of realization of irreducible representations

## Definition

### In characteristic zero

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

## Facts

- Linear representation is realizable over principal ideal domain iff it is realizable over field of fractions: In particular, if is a principal ideal domain, then any representation that can be realized over the field of fractions can also be realized over . 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