Minimal ring of realization of irreducible representations
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.
- 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