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## Main Question or Discussion Point

Matrix representation of a finite group G is irreducible representation if

[tex]\sum^n_{i=1}|\chi_i|^2=|G|[/tex].

Representation is reducible if

[tex]\sum^n_{i=1}|\chi_i|^2>|G|[/tex].

What if

[tex]\sum^n_{i=1}|\chi_i|^2<|G|[/tex].

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).

[tex]\sum^n_{i=1}|\chi_i|^2=|G|[/tex].

Representation is reducible if

[tex]\sum^n_{i=1}|\chi_i|^2>|G|[/tex].

What if

[tex]\sum^n_{i=1}|\chi_i|^2<|G|[/tex].

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).