Characters span class functions iff they separate conjugacy classes iff field contains field generated by character values

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Statement

Suppose G is a group and k is a field whose characteristic does not divide the order of G. The following are equivalent:

  1. The characters of irreducible representations of G over k form a basis for the space of class functions.
  2. The characters of irreducible representations of G over k span the space of class functions.
  3. The characters of all finite-dimensional linear representations of G over k span the space of class functions.
  4. Given any two distinct conjugacy classes of G, there is an irreducible representation whose character value is different on the two conjugacy classes.
  5. k contains a subfield K that is isomorphic to the field generated by character values in its characteristic, i.e., there is a splitting field containing k such that all irreducible representations of G over the splitting field have character values in k.

Note that k need not itself be a splitting field for G. 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).

Facts used

  1. Splitting implies characters form a basis for space of class functions, splitting implies characters separate conjugacy classes