Representations of finite groups: Irreducible and reducible

[LagrangeEuler]

Matrix representation of a finite group G is irreducible representation if
\sum^n_{i=1}|\chi_i|^2=|G|.
Representation is reducible if
\sum^n_{i=1}|\chi_i|^2>|G|.
What if
\sum^n_{i=1}|\chi_i|^2<|G|.
Are then multiplication of matrices form a group? If yes what we can say from $\sum^n_{i=1}|\chi_i|^2<|G|$. $\chi_i$ are characters (traces of matrices).

 

[Orodruin]

You cannot have that situation.