Real character values

The question of real-valued characters was briefly considered in Exercises 4 to 6 of Chapter 2.

In this section we are concerned with a particular element g of a finite group G and an arbitrary character ψ, which need not be irreducible. We have seen in (2.87) (p. 51) that

Hence we have

Proposition 6.1. The character value ψ(g) is real if and only if

We recall that all characters are class functions (Proposition 1.1(ii)): thus

if x and y are conjugate in G, which we write as

Hence if g–1 ∼ g, then equation (6.2) holds for every character ψ, and ψ(g) is therefore real. We shall now show that the converse is also true.

Theorem 6.1. The numbers ψ(g) are real for all characters ψ of a finite group if and only if

Proof. It only remains to prove that (6.3) is a necessary condition for the reality of all ψ(g). Suppose that g–1 and g belong to distinct conjugacy classes Cα and Cβ respectively. Put

Then by (6.1)

In particular, for each irreducible character χ(i) we have that

In this case the character relations of the second kind ((2.42), p. 51) imply that

It is therefore evident that not all the values of can be real if (6.3) is false.