Character Table - Outer Automorphisms

Outer Automorphisms

The outer automorphism group acts on the character table by permuting columns (conjugacy classes) and accordingly rows, which gives another symmetry to the table. For example, abelian groups have the outer automorphism which is non-trivial except for 2-groups, and outer because abelian groups are precisely those for which conjugation (inner automorphisms) acts trivially. In the example of above, this map sends and accordingly switches and (switching their values of and ). Note that this particular automorphism (negative in abelian groups) agrees with complex conjugation.

Formally, if is an automorphism of G and is a representation, then is a representation. If is an inner automorphism (conjugation by some element a), then it acts trivially on representations, because representations are class functions (conjugation does not change their value). Thus a given class of outer automorphisms, it acts on the characters – because inner automorphisms act trivially, the action of the automorphism group Aut descends to the quotient Out.

This relation can be used both ways: given an outer automorphism, one can produce new representations (if the representation is not equal on conjugacy classes that are interchanged by the outer automorphism), and conversely, one can restrict possible outer automorphisms based on the character table.