If
You could define this for any old functions on
Over the next few section, we will prove the following theorem-
Let
The irreducible characters are orthonormal with respect to the inner product defined above. This means that, if
We can also see this inner product as the standard inner product of the rows of the character table
with the entries weighted by
The number of irreducible representations is equal to the number of conjugacy classes. In other words, the character table is square.
The columns of the character table are orthonormal with respect to the weighted inner product from part(1).
That is, if
while if they are the same conjugacy class then
To complement this, we have:
Two irreducible representations of
In other words, each row of the character table corresponds to exactly one isomorphism class of irreducible representation.
A representation is determined up to isomorphism by its character.
If
Let the irreducible representations be
since
For the second part, note that
Since the