Central character value of a conjugacy-class sum
= Central character value of a conjugacy-class sum
For an irreducible character $\chi$ and a conjugacy class $C$ represented by $c$, the central group-algebra element $\sum_{x\in C}x$ acts as the scalar
$$
\omega_\chi(C)=\frac{|C|\chi(c)}{\chi(1)}.
$$
This scalar is an <algebraic integer>, since the class sum acts by an integer matrix on the regular representation.