Characters span class functions iff they separate conjugacy classes iff field contains field generated by character values
Suppose is a nontrivial group and is a field whose characteristic does not divide the order of . The following are equivalent:
- The characters of irreducible representations of over form a basis for the space of class functions.
- The characters of irreducible representations of over span the space of class functions.
- The characters of all finite-dimensional linear representations of over span the space of class functions.
- Given any two distinct conjugacy classes of , there is an irreducible representation whose character value is different on the two conjugacy classes.
- contains a subfield that is isomorphic to the field generated by character values in its characteristic, i.e., there is a splitting field containing such that all irreducible representations of over the splitting field have character values in .
Note that need not itself be a splitting field for . For instance, the field of rational numbers satisfies these equivalent conditions for the quaternion group (see linear representation theory of quaternion group, faithful irreducible representation of quaternion group).