Minimal ring of realization of irreducible representations

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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